id sid tid token lemma pos american_scientific_journal-8025 1 1 204 204 NUM american_scientific_journal-8025 1 2 american american ADJ american_scientific_journal-8025 1 3 academic academic ADJ american_scientific_journal-8025 1 4 scientific scientific ADJ american_scientific_journal-8025 1 5 research research NOUN american_scientific_journal-8025 1 6 journal journal NOUN american_scientific_journal-8025 1 7 for for ADP american_scientific_journal-8025 1 8 engineering engineering NOUN american_scientific_journal-8025 1 9 , , PUNCT american_scientific_journal-8025 1 10 technology technology NOUN american_scientific_journal-8025 1 11 , , PUNCT american_scientific_journal-8025 1 12 and and CCONJ american_scientific_journal-8025 1 13 sciences sciences PROPN american_scientific_journal-8025 1 14 issn issn PROPN american_scientific_journal-8025 1 15 ( ( PUNCT american_scientific_journal-8025 1 16 print print NOUN american_scientific_journal-8025 1 17 ) ) PUNCT american_scientific_journal-8025 1 18 2313 2313 NUM american_scientific_journal-8025 1 19 - - SYM american_scientific_journal-8025 1 20 4410 4410 NUM american_scientific_journal-8025 1 21 , , PUNCT american_scientific_journal-8025 1 22 issn issn PROPN american_scientific_journal-8025 1 23 ( ( PUNCT american_scientific_journal-8025 1 24 online online ADJ american_scientific_journal-8025 1 25 ) ) PUNCT american_scientific_journal-8025 1 26 2313 2313 NUM american_scientific_journal-8025 1 27 - - SYM american_scientific_journal-8025 1 28 4402 4402 NUM american_scientific_journal-8025 1 29 http://asrjetsjournal.org/ http://asrjetsjournal.org/ NOUN american_scientific_journal-8025 1 30 strong strong ADJ american_scientific_journal-8025 1 31 asymmetry asymmetry NOUN american_scientific_journal-8025 1 32 and and CCONJ american_scientific_journal-8025 1 33 the the DET american_scientific_journal-8025 1 34 mode mode NOUN american_scientific_journal-8025 1 35 , , PUNCT american_scientific_journal-8025 1 36 median median NOUN american_scientific_journal-8025 1 37 , , PUNCT american_scientific_journal-8025 1 38 and and CCONJ american_scientific_journal-8025 1 39 mean mean VERB american_scientific_journal-8025 1 40 inequality inequality PROPN american_scientific_journal-8025 1 41 j. j. PROPN american_scientific_journal-8025 1 42 gama gama PROPN american_scientific_journal-8025 1 43 a a PRON american_scientific_journal-8025 1 44 , , PUNCT american_scientific_journal-8025 1 45 c. c. PROPN american_scientific_journal-8025 1 46 e. e. PROPN american_scientific_journal-8025 1 47 suΓ‘rez suΓ‘rez PROPN american_scientific_journal-8025 2 1 b b PROPN american_scientific_journal-8025 2 2 * * PUNCT american_scientific_journal-8025 2 3 a a DET american_scientific_journal-8025 2 4 ufmg ufmg NOUN american_scientific_journal-8025 2 5 , , PUNCT american_scientific_journal-8025 2 6 av av PROPN american_scientific_journal-8025 2 7 . . PUNCT american_scientific_journal-8025 2 8 pres pres PROPN american_scientific_journal-8025 2 9 . . PUNCT american_scientific_journal-8025 2 10 antΓ΄nio antΓ΄nio PROPN american_scientific_journal-8025 2 11 carlos carlos PROPN american_scientific_journal-8025 2 12 , , PUNCT american_scientific_journal-8025 2 13 6627 6627 NUM american_scientific_journal-8025 2 14 , , PUNCT american_scientific_journal-8025 2 15 belo belo PROPN american_scientific_journal-8025 2 16 horizonte horizonte PROPN american_scientific_journal-8025 2 17 , , PUNCT american_scientific_journal-8025 2 18 brazil brazil PROPN american_scientific_journal-8025 2 19 b b PROPN american_scientific_journal-8025 2 20 impa impa PROPN american_scientific_journal-8025 2 21 , , PUNCT american_scientific_journal-8025 2 22 estrada estrada PROPN american_scientific_journal-8025 2 23 dona dona PROPN american_scientific_journal-8025 2 24 castorina castorina PROPN american_scientific_journal-8025 2 25 110 110 NUM american_scientific_journal-8025 2 26 , , PUNCT american_scientific_journal-8025 2 27 rio rio PROPN american_scientific_journal-8025 2 28 de de PROPN american_scientific_journal-8025 2 29 janeiro janeiro PROPN american_scientific_journal-8025 2 30 , , PUNCT american_scientific_journal-8025 2 31 brazil brazil PROPN american_scientific_journal-8025 2 32 a a DET american_scientific_journal-8025 2 33 email email NOUN american_scientific_journal-8025 2 34 : : PUNCT american_scientific_journal-8025 2 35 camplise@impa.br camplise@impa.br PROPN american_scientific_journal-8025 2 36 b b NOUN american_scientific_journal-8025 2 37 email email NOUN american_scientific_journal-8025 2 38 : : PUNCT american_scientific_journal-8025 3 1 gamatorres1000@gmail.com gamatorres1000@gmail.com X american_scientific_journal-8025 3 2 abstract abstract ADJ american_scientific_journal-8025 3 3 this this DET american_scientific_journal-8025 3 4 paper paper NOUN american_scientific_journal-8025 3 5 introduces introduce VERB american_scientific_journal-8025 3 6 the the DET american_scientific_journal-8025 3 7 consoled console VERB american_scientific_journal-8025 3 8 strong strong ADJ american_scientific_journal-8025 3 9 asymmetry asymmetry NOUN american_scientific_journal-8025 3 10 and and CCONJ american_scientific_journal-8025 3 11 variation variation NOUN american_scientific_journal-8025 3 12 coefficient coefficient NOUN american_scientific_journal-8025 3 13 as as ADP american_scientific_journal-8025 3 14 generalizations generalization NOUN american_scientific_journal-8025 3 15 of of ADP american_scientific_journal-8025 3 16 the the DET american_scientific_journal-8025 3 17 standard standard ADJ american_scientific_journal-8025 3 18 measures measure NOUN american_scientific_journal-8025 3 19 of of ADP american_scientific_journal-8025 3 20 pearson pearson PROPN american_scientific_journal-8025 3 21 skewness skewness PROPN american_scientific_journal-8025 3 22 coefficient coefficient NOUN american_scientific_journal-8025 3 23 and and CCONJ american_scientific_journal-8025 3 24 variance variance NOUN american_scientific_journal-8025 3 25 . . PUNCT american_scientific_journal-8025 4 1 using use VERB american_scientific_journal-8025 4 2 these these DET american_scientific_journal-8025 4 3 new new ADJ american_scientific_journal-8025 4 4 measures measure NOUN american_scientific_journal-8025 4 5 , , PUNCT american_scientific_journal-8025 4 6 the the DET american_scientific_journal-8025 4 7 concept concept NOUN american_scientific_journal-8025 4 8 of of ADP american_scientific_journal-8025 4 9 strong strong ADJ american_scientific_journal-8025 4 10 asymmetry asymmetry NOUN american_scientific_journal-8025 4 11 is be AUX american_scientific_journal-8025 4 12 introduced introduce VERB american_scientific_journal-8025 4 13 . . PUNCT american_scientific_journal-8025 5 1 we we PRON american_scientific_journal-8025 5 2 prove prove VERB american_scientific_journal-8025 5 3 that that SCONJ american_scientific_journal-8025 5 4 the the DET american_scientific_journal-8025 5 5 median median ADJ american_scientific_journal-8025 5 6 - - PUNCT american_scientific_journal-8025 5 7 mean mean NOUN american_scientific_journal-8025 5 8 inequality inequality NOUN american_scientific_journal-8025 5 9 is be AUX american_scientific_journal-8025 5 10 valid valid ADJ american_scientific_journal-8025 5 11 for for ADP american_scientific_journal-8025 5 12 that that DET american_scientific_journal-8025 5 13 kind kind NOUN american_scientific_journal-8025 5 14 of of ADP american_scientific_journal-8025 5 15 distribution distribution NOUN american_scientific_journal-8025 5 16 , , PUNCT american_scientific_journal-8025 5 17 but but CCONJ american_scientific_journal-8025 5 18 even even ADV american_scientific_journal-8025 5 19 in in ADP american_scientific_journal-8025 5 20 that that DET american_scientific_journal-8025 5 21 case case NOUN american_scientific_journal-8025 5 22 , , PUNCT american_scientific_journal-8025 5 23 there there PRON american_scientific_journal-8025 5 24 is be VERB american_scientific_journal-8025 5 25 no no DET american_scientific_journal-8025 5 26 relation relation NOUN american_scientific_journal-8025 5 27 between between ADP american_scientific_journal-8025 5 28 mode mode NOUN american_scientific_journal-8025 5 29 and and CCONJ american_scientific_journal-8025 5 30 median median NOUN american_scientific_journal-8025 5 31 . . PUNCT american_scientific_journal-8025 6 1 a a DET american_scientific_journal-8025 6 2 property property NOUN american_scientific_journal-8025 6 3 similar similar ADJ american_scientific_journal-8025 6 4 to to ADP american_scientific_journal-8025 6 5 first first ADJ american_scientific_journal-8025 6 6 - - PUNCT american_scientific_journal-8025 6 7 order order NOUN american_scientific_journal-8025 6 8 stochastic stochastic ADJ american_scientific_journal-8025 6 9 dominance dominance NOUN american_scientific_journal-8025 6 10 is be AUX american_scientific_journal-8025 6 11 proved prove VERB american_scientific_journal-8025 6 12 for for ADP american_scientific_journal-8025 6 13 the the DET american_scientific_journal-8025 6 14 variation variation NOUN american_scientific_journal-8025 6 15 coefficient coefficient NOUN american_scientific_journal-8025 6 16 . . PUNCT american_scientific_journal-8025 7 1 we we PRON american_scientific_journal-8025 7 2 also also ADV american_scientific_journal-8025 7 3 discuss discuss VERB american_scientific_journal-8025 7 4 the the DET american_scientific_journal-8025 7 5 implications implication NOUN american_scientific_journal-8025 7 6 of of ADP american_scientific_journal-8025 7 7 that that DET american_scientific_journal-8025 7 8 concept concept NOUN american_scientific_journal-8025 7 9 into into ADP american_scientific_journal-8025 7 10 economics economic NOUN american_scientific_journal-8025 7 11 . . PUNCT american_scientific_journal-8025 8 1 keywords keyword NOUN american_scientific_journal-8025 8 2 : : PUNCT american_scientific_journal-8025 8 3 skewness skewness NOUN american_scientific_journal-8025 8 4 ; ; PUNCT american_scientific_journal-8025 8 5 mode mode NOUN american_scientific_journal-8025 8 6 - - PUNCT american_scientific_journal-8025 8 7 median median NOUN american_scientific_journal-8025 8 8 - - PUNCT american_scientific_journal-8025 8 9 mean mean NOUN american_scientific_journal-8025 8 10 inequality inequality NOUN american_scientific_journal-8025 8 11 ; ; PUNCT american_scientific_journal-8025 8 12 measures measure NOUN american_scientific_journal-8025 8 13 of of ADP american_scientific_journal-8025 8 14 dispersion dispersion NOUN american_scientific_journal-8025 8 15 . . PUNCT american_scientific_journal-8025 9 1 1 1 X american_scientific_journal-8025 9 2 . . X american_scientific_journal-8025 9 3 introduction introduction NOUN american_scientific_journal-8025 9 4 there there PRON american_scientific_journal-8025 9 5 is be VERB american_scientific_journal-8025 9 6 a a DET american_scientific_journal-8025 9 7 false false ADJ american_scientific_journal-8025 9 8 belief belief NOUN american_scientific_journal-8025 9 9 that that SCONJ american_scientific_journal-8025 9 10 for for ADP american_scientific_journal-8025 9 11 every every DET american_scientific_journal-8025 9 12 unimodal unimodal ADJ american_scientific_journal-8025 9 13 and and CCONJ american_scientific_journal-8025 9 14 positive positive ADJ american_scientific_journal-8025 9 15 skewed skewed ADJ american_scientific_journal-8025 9 16 random random ADJ american_scientific_journal-8025 9 17 variable variable NOUN american_scientific_journal-8025 9 18 ( ( PUNCT american_scientific_journal-8025 9 19 that that PRON american_scientific_journal-8025 9 20 is be AUX american_scientific_journal-8025 9 21 𝐸(𝑋 𝐸(𝑋 NOUN american_scientific_journal-8025 9 22 βˆ’ βˆ’ PROPN american_scientific_journal-8025 9 23 𝐸𝑋)3 𝐸𝑋)3 X american_scientific_journal-8025 9 24 β‰₯ β‰₯ X american_scientific_journal-8025 9 25 0 0 NUM american_scientific_journal-8025 9 26 ) ) PUNCT american_scientific_journal-8025 9 27 , , PUNCT american_scientific_journal-8025 9 28 the the DET american_scientific_journal-8025 9 29 inequality inequality NOUN american_scientific_journal-8025 9 30 of of ADP american_scientific_journal-8025 9 31 π‘šπ‘œπ‘‘π‘’ π‘šπ‘œπ‘‘π‘’ ADJ american_scientific_journal-8025 9 32 ≀ ≀ NUM american_scientific_journal-8025 9 33 π‘šπ‘’π‘‘π‘–π‘Žπ‘› π‘šπ‘’π‘‘π‘–π‘Žπ‘› ADJ american_scientific_journal-8025 9 34 ≀ ≀ ADJ american_scientific_journal-8025 9 35 π‘šπ‘’π‘Žπ‘› π‘šπ‘’π‘Žπ‘› NOUN american_scientific_journal-8025 9 36 holds hold NOUN american_scientific_journal-8025 9 37 . . PUNCT american_scientific_journal-8025 10 1 several several ADJ american_scientific_journal-8025 10 2 authors author NOUN american_scientific_journal-8025 10 3 have have AUX american_scientific_journal-8025 10 4 shown show VERB american_scientific_journal-8025 10 5 that that SCONJ american_scientific_journal-8025 10 6 it it PRON american_scientific_journal-8025 10 7 is be AUX american_scientific_journal-8025 10 8 invalid invalid ADJ american_scientific_journal-8025 10 9 , , PUNCT american_scientific_journal-8025 10 10 and and CCONJ american_scientific_journal-8025 10 11 they they PRON american_scientific_journal-8025 10 12 have have AUX american_scientific_journal-8025 10 13 found find VERB american_scientific_journal-8025 10 14 conditions condition NOUN american_scientific_journal-8025 10 15 under under ADP american_scientific_journal-8025 10 16 which which PRON american_scientific_journal-8025 10 17 that that DET american_scientific_journal-8025 10 18 inequality inequality NOUN american_scientific_journal-8025 10 19 is be AUX american_scientific_journal-8025 10 20 valid valid ADJ american_scientific_journal-8025 10 21 . . PUNCT american_scientific_journal-8025 11 1 reference reference NOUN american_scientific_journal-8025 11 2 [ [ X american_scientific_journal-8025 11 3 1 1 X american_scientific_journal-8025 11 4 ] ] PUNCT american_scientific_journal-8025 11 5 showed show VERB american_scientific_journal-8025 11 6 a a DET american_scientific_journal-8025 11 7 list list NOUN american_scientific_journal-8025 11 8 of of ADP american_scientific_journal-8025 11 9 counterexamples counterexample NOUN american_scientific_journal-8025 11 10 for for ADP american_scientific_journal-8025 11 11 all all DET american_scientific_journal-8025 11 12 the the DET american_scientific_journal-8025 11 13 possible possible ADJ american_scientific_journal-8025 11 14 combinations combination NOUN american_scientific_journal-8025 11 15 of of ADP american_scientific_journal-8025 11 16 orders order NOUN american_scientific_journal-8025 11 17 between between ADP american_scientific_journal-8025 11 18 the the DET american_scientific_journal-8025 11 19 three three NUM american_scientific_journal-8025 11 20 measures measure NOUN american_scientific_journal-8025 11 21 . . PUNCT american_scientific_journal-8025 12 1 reference reference NOUN american_scientific_journal-8025 12 2 [ [ X american_scientific_journal-8025 12 3 11 11 NUM american_scientific_journal-8025 12 4 ] ] PUNCT american_scientific_journal-8025 12 5 found find VERB american_scientific_journal-8025 12 6 a a DET american_scientific_journal-8025 12 7 condition condition NOUN american_scientific_journal-8025 12 8 under under ADP american_scientific_journal-8025 12 9 which which PRON american_scientific_journal-8025 12 10 π‘šπ‘’π‘‘π‘–π‘Žπ‘› π‘šπ‘’π‘‘π‘–π‘Žπ‘› ADJ american_scientific_journal-8025 12 11 ≀ ≀ ADJ american_scientific_journal-8025 12 12 π‘šπ‘’π‘Žπ‘› π‘šπ‘’π‘Žπ‘› NOUN american_scientific_journal-8025 12 13 , , PUNCT american_scientific_journal-8025 12 14 but but CCONJ american_scientific_journal-8025 12 15 this this DET american_scientific_journal-8025 12 16 condition condition NOUN american_scientific_journal-8025 12 17 is be AUX american_scientific_journal-8025 12 18 not not PART american_scientific_journal-8025 12 19 related relate VERB american_scientific_journal-8025 12 20 to to ADP american_scientific_journal-8025 12 21 skewness skewness NOUN american_scientific_journal-8025 12 22 . . PUNCT american_scientific_journal-8025 13 1 instead instead ADV american_scientific_journal-8025 13 2 it it PRON american_scientific_journal-8025 13 3 is be AUX american_scientific_journal-8025 13 4 a a DET american_scientific_journal-8025 13 5 relation relation NOUN american_scientific_journal-8025 13 6 in in ADP american_scientific_journal-8025 13 7 the the DET american_scientific_journal-8025 13 8 cumulative cumulative ADJ american_scientific_journal-8025 13 9 distribution distribution NOUN american_scientific_journal-8025 13 10 function function NOUN american_scientific_journal-8025 13 11 . . PUNCT american_scientific_journal-8025 14 1 reference reference NOUN american_scientific_journal-8025 15 1 [ [ X american_scientific_journal-8025 15 2 2 2 X american_scientific_journal-8025 15 3 ] ] PUNCT american_scientific_journal-8025 15 4 have have VERB american_scientific_journal-8025 15 5 extensive extensive ADJ american_scientific_journal-8025 15 6 work work NOUN american_scientific_journal-8025 15 7 about about ADP american_scientific_journal-8025 15 8 the the DET american_scientific_journal-8025 15 9 class class NOUN american_scientific_journal-8025 15 10 of of ADP american_scientific_journal-8025 15 11 distributions distribution NOUN american_scientific_journal-8025 15 12 of of ADP american_scientific_journal-8025 15 13 fixed fix VERB american_scientific_journal-8025 15 14 variance variance NOUN american_scientific_journal-8025 15 15 and and CCONJ american_scientific_journal-8025 15 16 the the DET american_scientific_journal-8025 15 17 possible possible ADJ american_scientific_journal-8025 15 18 sorts sort NOUN american_scientific_journal-8025 15 19 in in ADP american_scientific_journal-8025 15 20 the the DET american_scientific_journal-8025 15 21 three three NUM american_scientific_journal-8025 15 22 central central ADJ american_scientific_journal-8025 15 23 measures measure NOUN american_scientific_journal-8025 15 24 . . PUNCT american_scientific_journal-8025 16 1 inspired inspire VERB american_scientific_journal-8025 16 2 by by ADP american_scientific_journal-8025 16 3 advances advance NOUN american_scientific_journal-8025 16 4 in in ADP american_scientific_journal-8025 16 5 economics economic NOUN american_scientific_journal-8025 16 6 such such ADJ american_scientific_journal-8025 16 7 as as ADP american_scientific_journal-8025 16 8 behavioral behavioral ADJ american_scientific_journal-8025 16 9 economics economic NOUN american_scientific_journal-8025 16 10 and and CCONJ american_scientific_journal-8025 16 11 specifically specifically ADV american_scientific_journal-8025 16 12 prospect prospect VERB american_scientific_journal-8025 16 13 theory theory NOUN american_scientific_journal-8025 16 14 , , PUNCT american_scientific_journal-8025 16 15 we we PRON american_scientific_journal-8025 16 16 have have AUX american_scientific_journal-8025 16 17 extended extend VERB american_scientific_journal-8025 16 18 the the DET american_scientific_journal-8025 16 19 classical classical ADJ american_scientific_journal-8025 16 20 measure measure NOUN american_scientific_journal-8025 16 21 of of ADP american_scientific_journal-8025 16 22 asymmetry asymmetry NOUN american_scientific_journal-8025 16 23 based base VERB american_scientific_journal-8025 16 24 on on ADP american_scientific_journal-8025 16 25 the the DET american_scientific_journal-8025 16 26 polynomial polynomial ADJ american_scientific_journal-8025 16 27 π‘₯3 π‘₯3 NOUN american_scientific_journal-8025 16 28 to to ADP american_scientific_journal-8025 16 29 any any DET american_scientific_journal-8025 16 30 odd odd ADJ american_scientific_journal-8025 16 31 and and CCONJ american_scientific_journal-8025 16 32 non non ADJ american_scientific_journal-8025 16 33 - - ADJ american_scientific_journal-8025 16 34 decreasing decrease VERB american_scientific_journal-8025 16 35 function function NOUN american_scientific_journal-8025 16 36 . . PUNCT american_scientific_journal-8025 17 1 using use VERB american_scientific_journal-8025 17 2 this this DET american_scientific_journal-8025 17 3 new new ADJ american_scientific_journal-8025 17 4 measure measure NOUN american_scientific_journal-8025 17 5 , , PUNCT american_scientific_journal-8025 17 6 we we PRON american_scientific_journal-8025 17 7 define define VERB american_scientific_journal-8025 17 8 a a DET american_scientific_journal-8025 17 9 class class NOUN american_scientific_journal-8025 17 10 of of ADP american_scientific_journal-8025 17 11 distributions distribution NOUN american_scientific_journal-8025 17 12 named name VERB american_scientific_journal-8025 17 13 strong strong ADJ american_scientific_journal-8025 17 14 asymmetric asymmetric NOUN american_scientific_journal-8025 17 15 , , PUNCT american_scientific_journal-8025 17 16 which which PRON american_scientific_journal-8025 17 17 have have VERB american_scientific_journal-8025 17 18 the the DET american_scientific_journal-8025 17 19 property property NOUN american_scientific_journal-8025 17 20 that that PRON american_scientific_journal-8025 17 21 its its PRON american_scientific_journal-8025 17 22 corresponding correspond VERB american_scientific_journal-8025 17 23 measure measure NOUN american_scientific_journal-8025 17 24 of of ADP american_scientific_journal-8025 17 25 skewness skewness NOUN american_scientific_journal-8025 17 26 relative relative ADJ american_scientific_journal-8025 17 27 to to ADP american_scientific_journal-8025 17 28 any any DET american_scientific_journal-8025 17 29 increasing increase VERB american_scientific_journal-8025 17 30 odd odd ADJ american_scientific_journal-8025 17 31 and and CCONJ american_scientific_journal-8025 17 32 convex convex ADJ american_scientific_journal-8025 17 33 function function NOUN american_scientific_journal-8025 17 34 in in ADP american_scientific_journal-8025 17 35 𝑅+ 𝑅+ PROPN american_scientific_journal-8025 17 36 is be AUX american_scientific_journal-8025 17 37 positive positive ADJ american_scientific_journal-8025 17 38 . . PUNCT american_scientific_journal-8025 18 1 the the DET american_scientific_journal-8025 18 2 inequality inequality NOUN american_scientific_journal-8025 18 3 π‘šπ‘’π‘‘π‘–π‘Žπ‘› π‘šπ‘’π‘‘π‘–π‘Žπ‘› NOUN american_scientific_journal-8025 18 4 βˆ’ βˆ’ PROPN american_scientific_journal-8025 18 5 π‘šπ‘’π‘Žπ‘› π‘šπ‘’π‘Žπ‘› NOUN american_scientific_journal-8025 18 6 and and CCONJ american_scientific_journal-8025 18 7 π‘šπ‘œπ‘‘π‘’ π‘šπ‘œπ‘‘π‘’ VERB american_scientific_journal-8025 18 8 βˆ’ βˆ’ PROPN american_scientific_journal-8025 18 9 π‘šπ‘’π‘Žπ‘› π‘šπ‘’π‘Žπ‘› NOUN american_scientific_journal-8025 18 10 can can AUX american_scientific_journal-8025 18 11 be be AUX american_scientific_journal-8025 18 12 established establish VERB american_scientific_journal-8025 18 13 for for ADP american_scientific_journal-8025 18 14 this this DET american_scientific_journal-8025 18 15 new new ADJ american_scientific_journal-8025 18 16 class class NOUN american_scientific_journal-8025 18 17 of of ADP american_scientific_journal-8025 18 18 functions function NOUN american_scientific_journal-8025 18 19 . . PUNCT american_scientific_journal-8025 19 1 similarly similarly ADV american_scientific_journal-8025 19 2 , , PUNCT american_scientific_journal-8025 19 3 a a DET american_scientific_journal-8025 19 4 generalization generalization NOUN american_scientific_journal-8025 19 5 of of ADP american_scientific_journal-8025 19 6 the the DET american_scientific_journal-8025 19 7 variance variance NOUN american_scientific_journal-8025 19 8 is be AUX american_scientific_journal-8025 19 9 shown show VERB american_scientific_journal-8025 19 10 and and CCONJ american_scientific_journal-8025 19 11 several several ADJ american_scientific_journal-8025 19 12 properties property NOUN american_scientific_journal-8025 19 13 related relate VERB american_scientific_journal-8025 19 14 to to ADP american_scientific_journal-8025 19 15 stochastic stochastic ADJ american_scientific_journal-8025 19 16 dominance dominance NOUN american_scientific_journal-8025 19 17 . . PUNCT american_scientific_journal-8025 20 1 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-8025 21 1 * * PUNCT american_scientific_journal-8025 21 2 corresponding correspond VERB american_scientific_journal-8025 21 3 author author NOUN american_scientific_journal-8025 21 4 . . PUNCT american_scientific_journal-8025 22 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-8025 22 2 american american PROPN american_scientific_journal-8025 22 3 academic academic PROPN american_scientific_journal-8025 22 4 scientific scientific ADJ american_scientific_journal-8025 22 5 research research NOUN american_scientific_journal-8025 22 6 journal journal NOUN american_scientific_journal-8025 22 7 for for ADP american_scientific_journal-8025 22 8 engineering engineering NOUN american_scientific_journal-8025 22 9 , , PUNCT american_scientific_journal-8025 22 10 technology technology NOUN american_scientific_journal-8025 22 11 , , PUNCT american_scientific_journal-8025 22 12 and and CCONJ american_scientific_journal-8025 22 13 sciences science NOUN american_scientific_journal-8025 22 14 ( ( PUNCT american_scientific_journal-8025 22 15 asrjets asrjet NOUN american_scientific_journal-8025 22 16 ) ) PUNCT american_scientific_journal-8025 22 17 ( ( PUNCT american_scientific_journal-8025 22 18 2022 2022 NUM american_scientific_journal-8025 22 19 ) ) PUNCT american_scientific_journal-8025 22 20 volume volume NOUN american_scientific_journal-8025 22 21 90 90 NUM american_scientific_journal-8025 22 22 , , PUNCT american_scientific_journal-8025 22 23 no1 no1 PROPN american_scientific_journal-8025 22 24 , , PUNCT american_scientific_journal-8025 22 25 pp pp ADV american_scientific_journal-8025 22 26 204 204 NUM american_scientific_journal-8025 22 27 - - SYM american_scientific_journal-8025 22 28 213 213 NUM american_scientific_journal-8025 22 29 205 205 NUM american_scientific_journal-8025 22 30 2 2 NUM american_scientific_journal-8025 22 31 . . PUNCT american_scientific_journal-8025 23 1 strong strong ADJ american_scientific_journal-8025 23 2 asymmetry asymmetry NOUN american_scientific_journal-8025 23 3 similar similar ADJ american_scientific_journal-8025 23 4 to to ADP american_scientific_journal-8025 23 5 some some DET american_scientific_journal-8025 23 6 notions notion NOUN american_scientific_journal-8025 23 7 introduced introduce VERB american_scientific_journal-8025 23 8 in in ADP american_scientific_journal-8025 23 9 [ [ X american_scientific_journal-8025 23 10 10,9 10,9 NUM american_scientific_journal-8025 23 11 ] ] X american_scientific_journal-8025 23 12 we we PRON american_scientific_journal-8025 23 13 make make VERB american_scientific_journal-8025 23 14 the the DET american_scientific_journal-8025 23 15 next next ADJ american_scientific_journal-8025 23 16 definitions definition NOUN american_scientific_journal-8025 23 17 . . PUNCT american_scientific_journal-8025 24 1 definition definition NOUN american_scientific_journal-8025 24 2 1 1 NUM american_scientific_journal-8025 24 3 : : PUNCT american_scientific_journal-8025 24 4 a a DET american_scientific_journal-8025 24 5 function function NOUN american_scientific_journal-8025 24 6 𝑓 𝑓 NOUN american_scientific_journal-8025 24 7 : : PUNCT american_scientific_journal-8025 24 8 𝑅 𝑅 PROPN american_scientific_journal-8025 24 9 β†’ β†’ SYM american_scientific_journal-8025 24 10 𝑅 𝑅 PROPN american_scientific_journal-8025 24 11 is be AUX american_scientific_journal-8025 24 12 called call VERB american_scientific_journal-8025 24 13 odd odd ADV american_scientific_journal-8025 24 14 - - PUNCT american_scientific_journal-8025 24 15 concave concave NOUN american_scientific_journal-8025 24 16 if if SCONJ american_scientific_journal-8025 24 17 it it PRON american_scientific_journal-8025 24 18 is be AUX american_scientific_journal-8025 24 19 odd odd ADJ american_scientific_journal-8025 24 20 and and CCONJ american_scientific_journal-8025 24 21 concave concave VERB american_scientific_journal-8025 24 22 in in ADP american_scientific_journal-8025 24 23 𝑅+ 𝑅+ PROPN american_scientific_journal-8025 24 24 . . PUNCT american_scientific_journal-8025 25 1 analogously analogously ADV american_scientific_journal-8025 25 2 , , PUNCT american_scientific_journal-8025 25 3 an an DET american_scientific_journal-8025 25 4 oddconvex oddconvex NOUN american_scientific_journal-8025 25 5 function function NOUN american_scientific_journal-8025 25 6 is be AUX american_scientific_journal-8025 25 7 odd odd ADJ american_scientific_journal-8025 25 8 and and CCONJ american_scientific_journal-8025 25 9 convex convex VERB american_scientific_journal-8025 25 10 in in ADP american_scientific_journal-8025 25 11 𝑅+ 𝑅+ PROPN american_scientific_journal-8025 25 12 definition definition NOUN american_scientific_journal-8025 25 13 2 2 NUM american_scientific_journal-8025 25 14 : : PUNCT american_scientific_journal-8025 25 15 let let VERB american_scientific_journal-8025 25 16 𝑋 𝑋 NOUN american_scientific_journal-8025 25 17 be be AUX american_scientific_journal-8025 25 18 a a DET american_scientific_journal-8025 25 19 random random ADJ american_scientific_journal-8025 25 20 variable variable NOUN american_scientific_journal-8025 25 21 and and CCONJ american_scientific_journal-8025 25 22 𝑓a 𝑓a NOUN american_scientific_journal-8025 25 23 continuous continuous ADJ american_scientific_journal-8025 25 24 , , PUNCT american_scientific_journal-8025 25 25 increasing increase VERB american_scientific_journal-8025 25 26 , , PUNCT american_scientific_journal-8025 25 27 and and CCONJ american_scientific_journal-8025 25 28 odd odd ADJ american_scientific_journal-8025 25 29 function function NOUN american_scientific_journal-8025 25 30 ; ; PUNCT american_scientific_journal-8025 25 31 we we PRON american_scientific_journal-8025 25 32 define define VERB american_scientific_journal-8025 25 33 ( ( PUNCT american_scientific_journal-8025 25 34 1 1 NUM american_scientific_journal-8025 25 35 ) ) PUNCT american_scientific_journal-8025 25 36 as as ADP american_scientific_journal-8025 25 37 the the DET american_scientific_journal-8025 25 38 skewness skewness NOUN american_scientific_journal-8025 25 39 coefficient coefficient NOUN american_scientific_journal-8025 25 40 of of ADP american_scientific_journal-8025 25 41 𝑋 𝑋 PROPN american_scientific_journal-8025 25 42 relative relative ADJ american_scientific_journal-8025 25 43 to to ADP american_scientific_journal-8025 25 44 the the DET american_scientific_journal-8025 25 45 function function NOUN american_scientific_journal-8025 25 46 𝑓 𝑓 ADP american_scientific_journal-8025 25 47 , , PUNCT american_scientific_journal-8025 25 48 and and CCONJ american_scientific_journal-8025 25 49 ( ( PUNCT american_scientific_journal-8025 25 50 2 2 X american_scientific_journal-8025 25 51 ) ) PUNCT american_scientific_journal-8025 25 52 as as ADP american_scientific_journal-8025 25 53 the the DET american_scientific_journal-8025 25 54 total total ADJ american_scientific_journal-8025 25 55 variation variation NOUN american_scientific_journal-8025 25 56 coefficient coefficient NOUN american_scientific_journal-8025 25 57 of of ADP american_scientific_journal-8025 25 58 π‘₯ π‘₯ PROPN american_scientific_journal-8025 25 59 relative relative ADJ american_scientific_journal-8025 25 60 to to ADP american_scientific_journal-8025 25 61 the the DET american_scientific_journal-8025 25 62 function function NOUN american_scientific_journal-8025 25 63 𝑓. 𝑓. NOUN american_scientific_journal-8025 25 64 remark remark NOUN american_scientific_journal-8025 25 65 3 3 NUM american_scientific_journal-8025 25 66 : : PUNCT american_scientific_journal-8025 25 67 suppose suppose VERB american_scientific_journal-8025 25 68 that that SCONJ american_scientific_journal-8025 25 69 𝑓 𝑓 PRON american_scientific_journal-8025 25 70 is be AUX american_scientific_journal-8025 25 71 a a DET american_scientific_journal-8025 25 72 continuous continuous ADJ american_scientific_journal-8025 25 73 , , PUNCT american_scientific_journal-8025 25 74 increasing increase VERB american_scientific_journal-8025 25 75 , , PUNCT american_scientific_journal-8025 25 76 and and CCONJ american_scientific_journal-8025 25 77 odd odd ADJ american_scientific_journal-8025 25 78 function function NOUN american_scientific_journal-8025 25 79 and and CCONJ american_scientific_journal-8025 25 80 𝑋 𝑋 NOUN american_scientific_journal-8025 25 81 is be AUX american_scientific_journal-8025 25 82 a a DET american_scientific_journal-8025 25 83 real real ADV american_scientific_journal-8025 25 84 random random ADJ american_scientific_journal-8025 25 85 variable variable NOUN american_scientific_journal-8025 25 86 with with ADP american_scientific_journal-8025 25 87 distribution distribution NOUN american_scientific_journal-8025 25 88 𝐹 𝐹 PROPN american_scientific_journal-8025 25 89 , , PUNCT american_scientific_journal-8025 25 90 such such ADJ american_scientific_journal-8025 25 91 that that DET american_scientific_journal-8025 25 92 𝐸[𝑋 𝐸[𝑋 NOUN american_scientific_journal-8025 25 93 ] ] PUNCT american_scientific_journal-8025 25 94 = = PUNCT american_scientific_journal-8025 26 1 0 0 X american_scientific_journal-8025 26 2 . . PUNCT american_scientific_journal-8025 27 1 the the DET american_scientific_journal-8025 27 2 skewness skewness NOUN american_scientific_journal-8025 27 3 of of ADP american_scientific_journal-8025 27 4 𝑋 𝑋 PROPN american_scientific_journal-8025 27 5 relative relative ADJ american_scientific_journal-8025 27 6 to to ADP american_scientific_journal-8025 27 7 𝑓 𝑓 PRON american_scientific_journal-8025 27 8 is be AUX american_scientific_journal-8025 27 9 ( ( PUNCT american_scientific_journal-8025 27 10 3 3 NUM american_scientific_journal-8025 27 11 ) ) PUNCT american_scientific_journal-8025 28 1 so so ADV american_scientific_journal-8025 28 2 , , PUNCT american_scientific_journal-8025 28 3 if if SCONJ american_scientific_journal-8025 28 4 π‘†π‘˜π‘“(𝑋 π‘†π‘˜π‘“(𝑋 NOUN american_scientific_journal-8025 28 5 ) ) PUNCT american_scientific_journal-8025 29 1 is be AUX american_scientific_journal-8025 29 2 positive positive ADJ american_scientific_journal-8025 29 3 , , PUNCT american_scientific_journal-8025 29 4 ( ( PUNCT american_scientific_journal-8025 29 5 4 4 X american_scientific_journal-8025 29 6 ) ) PUNCT american_scientific_journal-8025 29 7 the the DET american_scientific_journal-8025 29 8 right right ADJ american_scientific_journal-8025 29 9 side side NOUN american_scientific_journal-8025 29 10 of of ADP american_scientific_journal-8025 29 11 the the DET american_scientific_journal-8025 29 12 distribution distribution NOUN american_scientific_journal-8025 29 13 is be AUX american_scientific_journal-8025 29 14 under under ADP american_scientific_journal-8025 29 15 the the DET american_scientific_journal-8025 29 16 weights weight NOUN american_scientific_journal-8025 29 17 given give VERB american_scientific_journal-8025 29 18 by by ADP american_scientific_journal-8025 29 19 𝑓. 𝑓. NOUN american_scientific_journal-8025 29 20 ( ( PUNCT american_scientific_journal-8025 29 21 the the DET american_scientific_journal-8025 29 22 positive positive ADJ american_scientific_journal-8025 29 23 side side NOUN american_scientific_journal-8025 29 24 ) ) PUNCT american_scientific_journal-8025 30 1 is be AUX american_scientific_journal-8025 30 2 stronger strong ADJ american_scientific_journal-8025 30 3 than than ADP american_scientific_journal-8025 30 4 the the DET american_scientific_journal-8025 30 5 left left ADJ american_scientific_journal-8025 30 6 side side NOUN american_scientific_journal-8025 30 7 . . PUNCT american_scientific_journal-8025 31 1 suppose suppose VERB american_scientific_journal-8025 31 2 additionally additionally ADV american_scientific_journal-8025 31 3 that that SCONJ american_scientific_journal-8025 31 4 𝑓 𝑓 PRON american_scientific_journal-8025 31 5 is be AUX american_scientific_journal-8025 31 6 an an DET american_scientific_journal-8025 31 7 odd odd ADV american_scientific_journal-8025 31 8 - - PUNCT american_scientific_journal-8025 31 9 concave concave NOUN american_scientific_journal-8025 31 10 function function NOUN american_scientific_journal-8025 31 11 . . PUNCT american_scientific_journal-8025 32 1 therefore therefore ADV american_scientific_journal-8025 32 2 . . PUNCT american_scientific_journal-8025 33 1 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 33 2 ) ) PUNCT american_scientific_journal-8025 33 3 π‘₯ π‘₯ NOUN american_scientific_journal-8025 33 4 is be AUX american_scientific_journal-8025 33 5 a a DET american_scientific_journal-8025 33 6 decreasing decrease VERB american_scientific_journal-8025 33 7 function function NOUN american_scientific_journal-8025 33 8 , , PUNCT american_scientific_journal-8025 33 9 and and CCONJ american_scientific_journal-8025 33 10 π‘†π‘˜π‘“(𝑋 π‘†π‘˜π‘“(𝑋 NUM american_scientific_journal-8025 33 11 ) ) PUNCT american_scientific_journal-8025 33 12 is be AUX american_scientific_journal-8025 33 13 considerably considerably ADV american_scientific_journal-8025 33 14 more more ADV american_scientific_journal-8025 33 15 sensitive sensitive ADJ american_scientific_journal-8025 33 16 to to ADP american_scientific_journal-8025 33 17 values value NOUN american_scientific_journal-8025 33 18 close close ADV american_scientific_journal-8025 33 19 to to ADP american_scientific_journal-8025 33 20 zero zero NUM american_scientific_journal-8025 33 21 than than ADP american_scientific_journal-8025 33 22 to to ADP american_scientific_journal-8025 33 23 values value NOUN american_scientific_journal-8025 33 24 far far ADV american_scientific_journal-8025 33 25 from from ADP american_scientific_journal-8025 33 26 it it PRON american_scientific_journal-8025 33 27 , , PUNCT american_scientific_journal-8025 33 28 then then ADV american_scientific_journal-8025 33 29 we we PRON american_scientific_journal-8025 33 30 might might AUX american_scientific_journal-8025 33 31 expect expect VERB american_scientific_journal-8025 33 32 that that PRON american_scientific_journal-8025 33 33 𝐹′(π‘₯ 𝐹′(π‘₯ PUNCT american_scientific_journal-8025 33 34 ) ) PUNCT american_scientific_journal-8025 33 35 > > X american_scientific_journal-8025 34 1 𝐹′(βˆ’π‘₯ 𝐹′(βˆ’π‘₯ NUM american_scientific_journal-8025 34 2 ) ) PUNCT american_scientific_journal-8025 34 3 for for ADP american_scientific_journal-8025 34 4 values value NOUN american_scientific_journal-8025 34 5 near near ADP american_scientific_journal-8025 34 6 to to ADP american_scientific_journal-8025 34 7 0 0 NUM american_scientific_journal-8025 34 8 , , PUNCT american_scientific_journal-8025 34 9 i.e i.e PRON american_scientific_journal-8025 34 10 , , PUNCT american_scientific_journal-8025 34 11 intuitively intuitively ADV american_scientific_journal-8025 34 12 , , PUNCT american_scientific_journal-8025 34 13 the the DET american_scientific_journal-8025 34 14 distribution distribution NOUN american_scientific_journal-8025 34 15 is be AUX american_scientific_journal-8025 34 16 right right ADV american_scientific_journal-8025 34 17 skewed skewed ADJ american_scientific_journal-8025 34 18 when when SCONJ american_scientific_journal-8025 34 19 it it PRON american_scientific_journal-8025 34 20 is be AUX american_scientific_journal-8025 34 21 restricted restrict VERB american_scientific_journal-8025 34 22 to to ADP american_scientific_journal-8025 34 23 values value NOUN american_scientific_journal-8025 34 24 near near ADV american_scientific_journal-8025 34 25 to to ADP american_scientific_journal-8025 34 26 the the DET american_scientific_journal-8025 34 27 reference reference NOUN american_scientific_journal-8025 34 28 point point NOUN american_scientific_journal-8025 34 29 . . PUNCT american_scientific_journal-8025 35 1 however however ADV american_scientific_journal-8025 35 2 , , PUNCT american_scientific_journal-8025 35 3 since since SCONJ american_scientific_journal-8025 35 4 𝐹′(π‘₯ 𝐹′(π‘₯ NUM american_scientific_journal-8025 35 5 ) ) PUNCT american_scientific_journal-8025 35 6 > > X american_scientific_journal-8025 35 7 𝐹′(βˆ’π‘₯ 𝐹′(βˆ’π‘₯ NUM american_scientific_journal-8025 35 8 ) ) PUNCT american_scientific_journal-8025 35 9 for for ADP american_scientific_journal-8025 35 10 values value NOUN american_scientific_journal-8025 35 11 near near ADP american_scientific_journal-8025 35 12 to to ADP american_scientific_journal-8025 35 13 0 0 NUM american_scientific_journal-8025 35 14 we we PRON american_scientific_journal-8025 35 15 might might AUX american_scientific_journal-8025 35 16 have have VERB american_scientific_journal-8025 35 17 that that PRON american_scientific_journal-8025 35 18 , , PUNCT american_scientific_journal-8025 35 19 for for ADP american_scientific_journal-8025 35 20 values value NOUN american_scientific_journal-8025 35 21 far far ADV american_scientific_journal-8025 35 22 from from ADP american_scientific_journal-8025 35 23 0 0 NUM american_scientific_journal-8025 35 24 , , PUNCT american_scientific_journal-8025 35 25 𝐹′(π‘₯ 𝐹′(π‘₯ NUM american_scientific_journal-8025 35 26 ) ) PUNCT american_scientific_journal-8025 35 27 < < X american_scientific_journal-8025 35 28 𝐹′(βˆ’π‘₯ 𝐹′(βˆ’π‘₯ NUM american_scientific_journal-8025 35 29 ) ) PUNCT american_scientific_journal-8025 35 30 , , PUNCT american_scientific_journal-8025 35 31 which which PRON american_scientific_journal-8025 35 32 suggests suggest VERB american_scientific_journal-8025 35 33 that that SCONJ american_scientific_journal-8025 35 34 the the DET american_scientific_journal-8025 35 35 distribution distribution NOUN american_scientific_journal-8025 35 36 is be AUX american_scientific_journal-8025 35 37 left leave VERB american_scientific_journal-8025 35 38 - - PUNCT american_scientific_journal-8025 35 39 skewed skew VERB american_scientific_journal-8025 35 40 . . PUNCT american_scientific_journal-8025 36 1 thus thus ADV american_scientific_journal-8025 36 2 , , PUNCT american_scientific_journal-8025 36 3 a a DET american_scientific_journal-8025 36 4 positive positive ADJ american_scientific_journal-8025 36 5 𝑓 𝑓 DET american_scientific_journal-8025 36 6 skewness skewness NOUN american_scientific_journal-8025 36 7 coefficient coefficient NOUN american_scientific_journal-8025 36 8 with with ADP american_scientific_journal-8025 36 9 𝑓 𝑓 DET american_scientific_journal-8025 36 10 odd odd ADV american_scientific_journal-8025 36 11 - - PUNCT american_scientific_journal-8025 36 12 concave concave NOUN american_scientific_journal-8025 36 13 suggests suggest VERB american_scientific_journal-8025 36 14 that that SCONJ american_scientific_journal-8025 36 15 the the DET american_scientific_journal-8025 36 16 distribution distribution NOUN american_scientific_journal-8025 36 17 is be AUX american_scientific_journal-8025 36 18 left leave VERB american_scientific_journal-8025 36 19 - - PUNCT american_scientific_journal-8025 36 20 skewed skewed ADJ american_scientific_journal-8025 36 21 , , PUNCT american_scientific_journal-8025 36 22 and and CCONJ american_scientific_journal-8025 36 23 that that SCONJ american_scientific_journal-8025 36 24 π‘†π‘˜π‘”(𝐹 π‘†π‘˜π‘”(𝐹 X american_scientific_journal-8025 36 25 ) ) PUNCT american_scientific_journal-8025 36 26 might might AUX american_scientific_journal-8025 36 27 be be AUX american_scientific_journal-8025 36 28 negative negative ADJ american_scientific_journal-8025 36 29 if if SCONJ american_scientific_journal-8025 36 30 𝑔 𝑔 PROPN american_scientific_journal-8025 36 31 is be AUX american_scientific_journal-8025 36 32 a a DET american_scientific_journal-8025 36 33 odd odd ADJ american_scientific_journal-8025 36 34 - - PUNCT american_scientific_journal-8025 36 35 convex convex NOUN american_scientific_journal-8025 36 36 function function NOUN american_scientific_journal-8025 36 37 . . PUNCT american_scientific_journal-8025 37 1 on on ADP american_scientific_journal-8025 37 2 another another DET american_scientific_journal-8025 37 3 side side NOUN american_scientific_journal-8025 37 4 , , PUNCT american_scientific_journal-8025 37 5 if if SCONJ american_scientific_journal-8025 37 6 𝑋 𝑋 PROPN american_scientific_journal-8025 37 7 is be AUX american_scientific_journal-8025 37 8 a a DET american_scientific_journal-8025 37 9 symmetric symmetric ADJ american_scientific_journal-8025 37 10 random random ADJ american_scientific_journal-8025 37 11 variable variable NOUN american_scientific_journal-8025 37 12 βˆ€π‘₯ βˆ€π‘₯ NOUN american_scientific_journal-8025 37 13 , , PUNCT american_scientific_journal-8025 37 14 𝐹′(π‘₯ 𝐹′(π‘₯ NUM american_scientific_journal-8025 37 15 ) ) PUNCT american_scientific_journal-8025 37 16 = = SYM american_scientific_journal-8025 37 17 𝐹′(βˆ’π‘₯ 𝐹′(βˆ’π‘₯ NUM american_scientific_journal-8025 37 18 ) ) PUNCT american_scientific_journal-8025 37 19 π‘ π‘œ π‘ π‘œ ADP american_scientific_journal-8025 37 20 , , PUNCT american_scientific_journal-8025 37 21 π‘†π‘˜π‘“(𝑋 π‘†π‘˜π‘“(𝑋 NOUN american_scientific_journal-8025 37 22 ) ) PUNCT american_scientific_journal-8025 37 23 = = SYM american_scientific_journal-8025 37 24 0 0 X american_scientific_journal-8025 37 25 . . PUNCT american_scientific_journal-8025 37 26 remark remark NOUN american_scientific_journal-8025 37 27 4 4 NUM american_scientific_journal-8025 37 28 : : PUNCT american_scientific_journal-8025 37 29 some some DET american_scientific_journal-8025 37 30 properties property NOUN american_scientific_journal-8025 37 31 of of ADP american_scientific_journal-8025 37 32 the the DET american_scientific_journal-8025 37 33 skewness skewness NOUN american_scientific_journal-8025 37 34 coefficient coefficient NOUN american_scientific_journal-8025 37 35 relative relative ADJ american_scientific_journal-8025 37 36 to to ADP american_scientific_journal-8025 37 37 𝑓 𝑓 PRON american_scientific_journal-8025 37 38 are be AUX american_scientific_journal-8025 37 39 : : PUNCT american_scientific_journal-8025 37 40 american american ADJ american_scientific_journal-8025 37 41 academic academic ADJ american_scientific_journal-8025 37 42 scientific scientific ADJ american_scientific_journal-8025 37 43 research research NOUN american_scientific_journal-8025 37 44 journal journal NOUN american_scientific_journal-8025 37 45 for for ADP american_scientific_journal-8025 37 46 engineering engineering NOUN american_scientific_journal-8025 37 47 , , PUNCT american_scientific_journal-8025 37 48 technology technology NOUN american_scientific_journal-8025 37 49 , , PUNCT american_scientific_journal-8025 37 50 and and CCONJ american_scientific_journal-8025 37 51 sciences science NOUN american_scientific_journal-8025 37 52 ( ( PUNCT american_scientific_journal-8025 37 53 asrjets asrjet NOUN american_scientific_journal-8025 37 54 ) ) PUNCT american_scientific_journal-8025 37 55 ( ( PUNCT american_scientific_journal-8025 37 56 2022 2022 NUM american_scientific_journal-8025 37 57 ) ) PUNCT american_scientific_journal-8025 37 58 volume volume NOUN american_scientific_journal-8025 37 59 90 90 NUM american_scientific_journal-8025 37 60 , , PUNCT american_scientific_journal-8025 37 61 no1 no1 PROPN american_scientific_journal-8025 37 62 , , PUNCT american_scientific_journal-8025 37 63 pp pp ADV american_scientific_journal-8025 37 64 204 204 NUM american_scientific_journal-8025 37 65 - - SYM american_scientific_journal-8025 37 66 213 213 NUM american_scientific_journal-8025 37 67 206 206 NUM american_scientific_journal-8025 37 68 1 1 NUM american_scientific_journal-8025 37 69 . . PUNCT american_scientific_journal-8025 38 1 if if SCONJ american_scientific_journal-8025 38 2 𝐹 𝐹 PROPN american_scientific_journal-8025 38 3 has have VERB american_scientific_journal-8025 38 4 a a DET american_scientific_journal-8025 38 5 symmetric symmetric ADJ american_scientific_journal-8025 38 6 distribution distribution NOUN american_scientific_journal-8025 38 7 then then ADV american_scientific_journal-8025 38 8 π‘†π‘˜π‘“(𝐹 π‘†π‘˜π‘“(𝐹 NOUN american_scientific_journal-8025 38 9 ) ) PUNCT american_scientific_journal-8025 38 10 = = SYM american_scientific_journal-8025 38 11 0 0 NUM american_scientific_journal-8025 38 12 for for ADP american_scientific_journal-8025 38 13 every every DET american_scientific_journal-8025 38 14 odd odd ADJ american_scientific_journal-8025 38 15 function function NOUN american_scientific_journal-8025 38 16 𝑓 𝑓 X american_scientific_journal-8025 38 17 . . PUNCT american_scientific_journal-8025 39 1 if if SCONJ american_scientific_journal-8025 39 2 for for ADP american_scientific_journal-8025 39 3 every every DET american_scientific_journal-8025 39 4 odd odd ADJ american_scientific_journal-8025 39 5 and and CCONJ american_scientific_journal-8025 39 6 increasing increase VERB american_scientific_journal-8025 39 7 function function NOUN american_scientific_journal-8025 39 8 𝑓 𝑓 ADP american_scientific_journal-8025 39 9 , , PUNCT american_scientific_journal-8025 39 10 π‘†π‘˜π‘“(𝑋 π‘†π‘˜π‘“(𝑋 NOUN american_scientific_journal-8025 39 11 ) ) PUNCT american_scientific_journal-8025 39 12 = = SYM american_scientific_journal-8025 39 13 0 0 NUM american_scientific_journal-8025 39 14 then then ADV american_scientific_journal-8025 39 15 𝑋 𝑋 PROPN american_scientific_journal-8025 39 16 is be AUX american_scientific_journal-8025 39 17 symmetric symmetric ADJ american_scientific_journal-8025 39 18 . . PUNCT american_scientific_journal-8025 40 1 2 2 X american_scientific_journal-8025 40 2 . . X american_scientific_journal-8025 40 3 if if SCONJ american_scientific_journal-8025 40 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 40 5 ) ) PUNCT american_scientific_journal-8025 40 6 = = SYM american_scientific_journal-8025 40 7 π‘₯3 π‘₯3 NOUN american_scientific_journal-8025 40 8 then then ADV american_scientific_journal-8025 40 9 π‘†π‘˜π‘“(π‘₯ π‘†π‘˜π‘“(π‘₯ AUX american_scientific_journal-8025 40 10 ) ) PUNCT american_scientific_journal-8025 40 11 correspond correspond VERB american_scientific_journal-8025 40 12 with with ADP american_scientific_journal-8025 40 13 the the DET american_scientific_journal-8025 40 14 third third ADJ american_scientific_journal-8025 40 15 moment moment NOUN american_scientific_journal-8025 40 16 around around ADP american_scientific_journal-8025 40 17 the the DET american_scientific_journal-8025 40 18 mean mean NOUN american_scientific_journal-8025 40 19 . . PUNCT american_scientific_journal-8025 41 1 3 3 X american_scientific_journal-8025 41 2 . . X american_scientific_journal-8025 41 3 if if SCONJ american_scientific_journal-8025 41 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 41 5 ) ) PUNCT american_scientific_journal-8025 41 6 = = SYM american_scientific_journal-8025 41 7 πœ‚π‘₯ πœ‚π‘₯ VERB american_scientific_journal-8025 41 8 then then ADV american_scientific_journal-8025 41 9 for for ADP american_scientific_journal-8025 41 10 all all PRON american_scientific_journal-8025 41 11 π‘₯ π‘₯ PROPN american_scientific_journal-8025 41 12 , , PUNCT american_scientific_journal-8025 41 13 π‘†π‘˜π‘“(π‘₯ π‘†π‘˜π‘“(π‘₯ NOUN american_scientific_journal-8025 41 14 ) ) PUNCT american_scientific_journal-8025 41 15 = = SYM american_scientific_journal-8025 41 16 0 0 X american_scientific_journal-8025 41 17 . . NOUN american_scientific_journal-8025 41 18 example example NOUN american_scientific_journal-8025 41 19 5 5 NUM american_scientific_journal-8025 41 20 : : PUNCT american_scientific_journal-8025 41 21 consider consider VERB american_scientific_journal-8025 41 22 𝐹 𝐹 PRON american_scientific_journal-8025 41 23 the the DET american_scientific_journal-8025 41 24 lottery lottery NOUN american_scientific_journal-8025 41 25 given give VERB american_scientific_journal-8025 41 26 by by ADP american_scientific_journal-8025 41 27 ( ( PUNCT american_scientific_journal-8025 41 28 βˆ’πΏ βˆ’πΏ PROPN american_scientific_journal-8025 41 29 , , PUNCT american_scientific_journal-8025 41 30 1 1 NUM american_scientific_journal-8025 41 31 βˆ’ βˆ’ PROPN american_scientific_journal-8025 41 32 𝑝 𝑝 PROPN american_scientific_journal-8025 41 33 ; ; PUNCT american_scientific_journal-8025 41 34 𝐺 𝐺 PROPN american_scientific_journal-8025 41 35 , , PUNCT american_scientific_journal-8025 41 36 𝑝 𝑝 NOUN american_scientific_journal-8025 41 37 ) ) PUNCT american_scientific_journal-8025 41 38 with with ADP american_scientific_journal-8025 41 39 𝐺 𝐺 PROPN american_scientific_journal-8025 41 40 , , PUNCT american_scientific_journal-8025 41 41 𝐿 𝐿 PROPN american_scientific_journal-8025 41 42 > > X american_scientific_journal-8025 41 43 0 0 NUM american_scientific_journal-8025 41 44 , , PUNCT american_scientific_journal-8025 41 45 that that ADV american_scientific_journal-8025 41 46 is is ADV american_scientific_journal-8025 41 47 , , PUNCT american_scientific_journal-8025 41 48 there there PRON american_scientific_journal-8025 41 49 is be VERB american_scientific_journal-8025 41 50 a a DET american_scientific_journal-8025 41 51 probability probability NOUN american_scientific_journal-8025 41 52 𝑝 𝑝 NOUN american_scientific_journal-8025 41 53 to to PART american_scientific_journal-8025 41 54 gain gain VERB american_scientific_journal-8025 41 55 𝐺 𝐺 NOUN american_scientific_journal-8025 41 56 and and CCONJ american_scientific_journal-8025 41 57 probability probability NOUN american_scientific_journal-8025 41 58 1 1 NUM american_scientific_journal-8025 41 59 βˆ’ βˆ’ PROPN american_scientific_journal-8025 41 60 𝑝 𝑝 NOUN american_scientific_journal-8025 41 61 of of ADP american_scientific_journal-8025 41 62 loss loss NOUN american_scientific_journal-8025 41 63 l. l. PROPN american_scientific_journal-8025 41 64 suppose suppose VERB american_scientific_journal-8025 41 65 that that SCONJ american_scientific_journal-8025 41 66 𝐸(𝐿 𝐸(𝐿 NOUN american_scientific_journal-8025 41 67 ) ) PUNCT american_scientific_journal-8025 42 1 = = PUNCT american_scientific_journal-8025 42 2 0 0 PUNCT american_scientific_journal-8025 43 1 so so ADV american_scientific_journal-8025 43 2 𝑝 𝑝 NOUN american_scientific_journal-8025 43 3 = = SYM american_scientific_journal-8025 43 4 𝐿 𝐿 PROPN american_scientific_journal-8025 43 5 𝐺+𝐿 𝐺+𝐿 ADJ american_scientific_journal-8025 43 6 and and CCONJ american_scientific_journal-8025 43 7 1 1 NUM american_scientific_journal-8025 43 8 βˆ’ βˆ’ PROPN american_scientific_journal-8025 43 9 𝑝 𝑝 NOUN american_scientific_journal-8025 43 10 = = SYM american_scientific_journal-8025 43 11 𝐺 𝐺 PROPN american_scientific_journal-8025 43 12 𝐺+𝐿 𝐺+𝐿 NOUN american_scientific_journal-8025 43 13 . . PUNCT american_scientific_journal-8025 44 1 let let VERB american_scientific_journal-8025 44 2 𝑓 𝑓 PRON american_scientific_journal-8025 44 3 strictly strictly ADV american_scientific_journal-8025 44 4 oddconvex oddconvex VERB american_scientific_journal-8025 44 5 ( ( PUNCT american_scientific_journal-8025 44 6 5 5 NUM american_scientific_journal-8025 44 7 ) ) PUNCT american_scientific_journal-8025 44 8 then then ADV american_scientific_journal-8025 44 9 π‘†π‘˜π‘“(𝐹 π‘†π‘˜π‘“(𝐹 X american_scientific_journal-8025 44 10 ) ) PUNCT american_scientific_journal-8025 44 11 > > X american_scientific_journal-8025 44 12 0 0 NUM american_scientific_journal-8025 45 1 iff iff PROPN american_scientific_journal-8025 45 2 ( ( PUNCT american_scientific_journal-8025 45 3 6 6 NUM american_scientific_journal-8025 45 4 ) ) PUNCT american_scientific_journal-8025 45 5 because because SCONJ american_scientific_journal-8025 45 6 𝑓(0 𝑓(0 NOUN american_scientific_journal-8025 45 7 ) ) PUNCT american_scientific_journal-8025 45 8 = = SYM american_scientific_journal-8025 45 9 0 0 PUNCT american_scientific_journal-8025 45 10 and and CCONJ american_scientific_journal-8025 45 11 𝑓 𝑓 PRON american_scientific_journal-8025 45 12 is be AUX american_scientific_journal-8025 45 13 strictly strictly ADV american_scientific_journal-8025 45 14 odd odd ADJ american_scientific_journal-8025 45 15 - - PUNCT american_scientific_journal-8025 45 16 convex convex NOUN american_scientific_journal-8025 45 17 , , PUNCT american_scientific_journal-8025 45 18 the the DET american_scientific_journal-8025 45 19 last last ADJ american_scientific_journal-8025 45 20 relation relation NOUN american_scientific_journal-8025 45 21 is be AUX american_scientific_journal-8025 45 22 valid valid ADJ american_scientific_journal-8025 45 23 iff iff PROPN american_scientific_journal-8025 45 24 𝐺 𝐺 PROPN american_scientific_journal-8025 45 25 > > X american_scientific_journal-8025 45 26 𝐿. 𝐿. VERB american_scientific_journal-8025 45 27 analogously analogously ADV american_scientific_journal-8025 45 28 if if SCONJ american_scientific_journal-8025 45 29 𝑓 𝑓 PRON american_scientific_journal-8025 45 30 is be AUX american_scientific_journal-8025 45 31 strictly strictly ADV american_scientific_journal-8025 45 32 odd odd ADV american_scientific_journal-8025 45 33 - - PUNCT american_scientific_journal-8025 45 34 concave concave ADJ american_scientific_journal-8025 45 35 π‘†π‘˜π‘“(𝐹 π‘†π‘˜π‘“(𝐹 NOUN american_scientific_journal-8025 45 36 ) ) PUNCT american_scientific_journal-8025 45 37 > > X american_scientific_journal-8025 45 38 0 0 PUNCT american_scientific_journal-8025 46 1 if if SCONJ american_scientific_journal-8025 46 2 𝐺 𝐺 PROPN american_scientific_journal-8025 46 3 < < X american_scientific_journal-8025 46 4 𝐿. 𝐿. VERB american_scientific_journal-8025 46 5 the the DET american_scientific_journal-8025 46 6 last last ADJ american_scientific_journal-8025 46 7 example example NOUN american_scientific_journal-8025 46 8 inspires inspire VERB american_scientific_journal-8025 46 9 the the DET american_scientific_journal-8025 46 10 following follow VERB american_scientific_journal-8025 46 11 definition definition NOUN american_scientific_journal-8025 46 12 : : PUNCT american_scientific_journal-8025 46 13 definition definition NOUN american_scientific_journal-8025 46 14 6 6 NUM american_scientific_journal-8025 46 15 : : PUNCT american_scientific_journal-8025 46 16 a a DET american_scientific_journal-8025 46 17 random random ADJ american_scientific_journal-8025 46 18 variable variable NOUN american_scientific_journal-8025 46 19 𝑋 𝑋 NOUN american_scientific_journal-8025 46 20 is be AUX american_scientific_journal-8025 46 21 strongly strongly ADV american_scientific_journal-8025 46 22 asymmetric asymmetric ADJ american_scientific_journal-8025 46 23 to to ADP american_scientific_journal-8025 46 24 the the DET american_scientific_journal-8025 46 25 right right NOUN american_scientific_journal-8025 46 26 if if SCONJ american_scientific_journal-8025 46 27 for for ADP american_scientific_journal-8025 46 28 all all DET american_scientific_journal-8025 46 29 𝑓 𝑓 DET american_scientific_journal-8025 46 30 increasing increase VERB american_scientific_journal-8025 46 31 and and CCONJ american_scientific_journal-8025 46 32 odd odd ADJ american_scientific_journal-8025 46 33 - - PUNCT american_scientific_journal-8025 46 34 convex convex NOUN american_scientific_journal-8025 46 35 , , PUNCT american_scientific_journal-8025 46 36 π‘†π‘˜π‘“(𝑋 π‘†π‘˜π‘“(𝑋 NOUN american_scientific_journal-8025 46 37 ) ) PUNCT american_scientific_journal-8025 46 38 β‰₯ β‰₯ NOUN american_scientific_journal-8025 46 39 0 0 NUM american_scientific_journal-8025 46 40 . . PUNCT american_scientific_journal-8025 47 1 analogously analogously ADV american_scientific_journal-8025 47 2 𝑋 𝑋 PROPN american_scientific_journal-8025 47 3 is be AUX american_scientific_journal-8025 47 4 strongly strongly ADV american_scientific_journal-8025 47 5 asymmetric asymmetric ADJ american_scientific_journal-8025 47 6 to to ADP american_scientific_journal-8025 47 7 the the DET american_scientific_journal-8025 47 8 left left NOUN american_scientific_journal-8025 47 9 if if SCONJ american_scientific_journal-8025 47 10 for for ADP american_scientific_journal-8025 47 11 all all DET american_scientific_journal-8025 47 12 𝑓 𝑓 DET american_scientific_journal-8025 47 13 increasing increase VERB american_scientific_journal-8025 47 14 and and CCONJ american_scientific_journal-8025 47 15 odd odd ADJ american_scientific_journal-8025 47 16 - - PUNCT american_scientific_journal-8025 47 17 convex convex NOUN american_scientific_journal-8025 47 18 , , PUNCT american_scientific_journal-8025 47 19 π‘†π‘˜π‘“(𝑋 π‘†π‘˜π‘“(𝑋 NOUN american_scientific_journal-8025 47 20 ) ) PUNCT american_scientific_journal-8025 47 21 ≀ ≀ NUM american_scientific_journal-8025 47 22 0 0 NUM american_scientific_journal-8025 47 23 . . PUNCT american_scientific_journal-8025 47 24 example example NOUN american_scientific_journal-8025 48 1 7 7 NUM american_scientific_journal-8025 48 2 : : PUNCT american_scientific_journal-8025 48 3 let let VERB american_scientific_journal-8025 48 4 𝑋 𝑋 PROPN american_scientific_journal-8025 48 5 a a DET american_scientific_journal-8025 48 6 random random ADJ american_scientific_journal-8025 48 7 variable variable NOUN american_scientific_journal-8025 48 8 with with ADP american_scientific_journal-8025 48 9 exponential exponential ADJ american_scientific_journal-8025 48 10 distribution distribution NOUN american_scientific_journal-8025 48 11 and and CCONJ american_scientific_journal-8025 48 12 density density NOUN american_scientific_journal-8025 48 13 function function NOUN american_scientific_journal-8025 48 14 is be AUX american_scientific_journal-8025 48 15 given give VERB american_scientific_journal-8025 48 16 by by ADP american_scientific_journal-8025 48 17 ( ( PUNCT american_scientific_journal-8025 48 18 7 7 X american_scientific_journal-8025 48 19 ) ) PUNCT american_scientific_journal-8025 48 20 this this DET american_scientific_journal-8025 48 21 distribution distribution NOUN american_scientific_journal-8025 48 22 is be AUX american_scientific_journal-8025 48 23 strongly strongly ADV american_scientific_journal-8025 48 24 asymmetric asymmetric ADJ american_scientific_journal-8025 48 25 to to ADP american_scientific_journal-8025 48 26 the the DET american_scientific_journal-8025 48 27 right right NOUN american_scientific_journal-8025 48 28 . . PUNCT american_scientific_journal-8025 49 1 let let VERB american_scientific_journal-8025 49 2 𝑓 𝑓 PRON american_scientific_journal-8025 49 3 be be AUX american_scientific_journal-8025 49 4 a a DET american_scientific_journal-8025 49 5 continuous continuous ADJ american_scientific_journal-8025 49 6 and and CCONJ american_scientific_journal-8025 49 7 odd odd ADJ american_scientific_journal-8025 49 8 - - PUNCT american_scientific_journal-8025 49 9 convex convex NOUN american_scientific_journal-8025 49 10 function function NOUN american_scientific_journal-8025 49 11 then then ADV american_scientific_journal-8025 49 12 ( ( PUNCT american_scientific_journal-8025 49 13 8) 8) NUM american_scientific_journal-8025 49 14 because because SCONJ american_scientific_journal-8025 49 15 of of ADP american_scientific_journal-8025 49 16 the the DET american_scientific_journal-8025 49 17 convexity convexity NOUN american_scientific_journal-8025 49 18 of of ADP american_scientific_journal-8025 49 19 𝑓 𝑓 PROPN american_scientific_journal-8025 49 20 , , PUNCT american_scientific_journal-8025 49 21 for for ADP american_scientific_journal-8025 49 22 each each DET american_scientific_journal-8025 49 23 0 0 NUM american_scientific_journal-8025 49 24 ≀ ≀ NOUN american_scientific_journal-8025 49 25 𝑦 𝑦 NOUN american_scientific_journal-8025 49 26 ≀ ≀ NUM american_scientific_journal-8025 49 27 1 1 NUM american_scientific_journal-8025 49 28 , , PUNCT american_scientific_journal-8025 49 29 𝑓(𝑦 𝑓(𝑦 ADV american_scientific_journal-8025 49 30 ) ) PUNCT american_scientific_journal-8025 49 31 ≀ ≀ NOUN american_scientific_journal-8025 50 1 𝑦𝑓(1 𝑦𝑓(1 NOUN american_scientific_journal-8025 50 2 ) ) PUNCT american_scientific_journal-8025 50 3 , , PUNCT american_scientific_journal-8025 50 4 we we PRON american_scientific_journal-8025 50 5 have have VERB american_scientific_journal-8025 50 6 that that DET american_scientific_journal-8025 50 7 american american ADJ american_scientific_journal-8025 50 8 academic academic ADJ american_scientific_journal-8025 50 9 scientific scientific ADJ american_scientific_journal-8025 50 10 research research NOUN american_scientific_journal-8025 50 11 journal journal NOUN american_scientific_journal-8025 50 12 for for ADP american_scientific_journal-8025 50 13 engineering engineering NOUN american_scientific_journal-8025 50 14 , , PUNCT american_scientific_journal-8025 50 15 technology technology NOUN american_scientific_journal-8025 50 16 , , PUNCT american_scientific_journal-8025 50 17 and and CCONJ american_scientific_journal-8025 50 18 sciences science NOUN american_scientific_journal-8025 50 19 ( ( PUNCT american_scientific_journal-8025 50 20 asrjets asrjet NOUN american_scientific_journal-8025 50 21 ) ) PUNCT american_scientific_journal-8025 50 22 ( ( PUNCT american_scientific_journal-8025 50 23 2022 2022 NUM american_scientific_journal-8025 50 24 ) ) PUNCT american_scientific_journal-8025 50 25 volume volume NOUN american_scientific_journal-8025 50 26 90 90 NUM american_scientific_journal-8025 50 27 , , PUNCT american_scientific_journal-8025 50 28 no1 no1 PROPN american_scientific_journal-8025 50 29 , , PUNCT american_scientific_journal-8025 50 30 pp pp ADV american_scientific_journal-8025 50 31 204 204 NUM american_scientific_journal-8025 50 32 - - SYM american_scientific_journal-8025 50 33 213 213 NUM american_scientific_journal-8025 50 34 207 207 NUM american_scientific_journal-8025 50 35 ( ( PUNCT american_scientific_journal-8025 50 36 9 9 NUM american_scientific_journal-8025 50 37 ) ) PUNCT american_scientific_journal-8025 50 38 on on ADP american_scientific_journal-8025 50 39 other other ADJ american_scientific_journal-8025 50 40 side side NOUN american_scientific_journal-8025 50 41 for for ADP american_scientific_journal-8025 50 42 each each DET american_scientific_journal-8025 50 43 𝑦 𝑦 NOUN american_scientific_journal-8025 50 44 > > X american_scientific_journal-8025 50 45 1 1 NUM american_scientific_journal-8025 50 46 , , PUNCT american_scientific_journal-8025 50 47 𝑓(𝑦 𝑓(𝑦 PROPN american_scientific_journal-8025 50 48 ) ) PUNCT american_scientific_journal-8025 50 49 > > X american_scientific_journal-8025 51 1 𝑓′(1)(𝑦 𝑓′(1)(𝑦 NOUN american_scientific_journal-8025 51 2 βˆ’ βˆ’ ADP american_scientific_journal-8025 51 3 1 1 NUM american_scientific_journal-8025 51 4 ) ) PUNCT american_scientific_journal-8025 51 5 + + CCONJ american_scientific_journal-8025 51 6 𝑓(1 𝑓(1 NOUN american_scientific_journal-8025 51 7 ) ) PUNCT american_scientific_journal-8025 51 8 , , PUNCT american_scientific_journal-8025 51 9 and and CCONJ american_scientific_journal-8025 51 10 we we PRON american_scientific_journal-8025 51 11 have have VERB american_scientific_journal-8025 51 12 that that SCONJ american_scientific_journal-8025 51 13 therefore therefore ADV american_scientific_journal-8025 51 14 , , PUNCT american_scientific_journal-8025 51 15 we we PRON american_scientific_journal-8025 51 16 have have VERB american_scientific_journal-8025 51 17 that that PRON american_scientific_journal-8025 51 18 . . PUNCT american_scientific_journal-8025 52 1 ( ( PUNCT american_scientific_journal-8025 52 2 10 10 NUM american_scientific_journal-8025 52 3 ) ) PUNCT american_scientific_journal-8025 52 4 in in ADP american_scientific_journal-8025 52 5 particular particular ADJ american_scientific_journal-8025 52 6 , , PUNCT american_scientific_journal-8025 52 7 note note VERB american_scientific_journal-8025 52 8 that that SCONJ american_scientific_journal-8025 52 9 if if SCONJ american_scientific_journal-8025 52 10 𝑓 𝑓 PRON american_scientific_journal-8025 52 11 is be AUX american_scientific_journal-8025 52 12 strictly strictly ADV american_scientific_journal-8025 52 13 odd odd ADJ american_scientific_journal-8025 52 14 - - PUNCT american_scientific_journal-8025 52 15 convex convex NOUN american_scientific_journal-8025 52 16 , , PUNCT american_scientific_journal-8025 52 17 the the DET american_scientific_journal-8025 52 18 inequalities inequality NOUN american_scientific_journal-8025 52 19 above above ADV american_scientific_journal-8025 52 20 are be AUX american_scientific_journal-8025 52 21 strict strict ADJ american_scientific_journal-8025 52 22 . . PUNCT american_scientific_journal-8025 53 1 motivated motivate VERB american_scientific_journal-8025 53 2 by by ADP american_scientific_journal-8025 53 3 example example NOUN american_scientific_journal-8025 53 4 7 7 NUM american_scientific_journal-8025 53 5 , , PUNCT american_scientific_journal-8025 53 6 we we PRON american_scientific_journal-8025 53 7 have have VERB american_scientific_journal-8025 53 8 the the DET american_scientific_journal-8025 53 9 following follow VERB american_scientific_journal-8025 53 10 result result NOUN american_scientific_journal-8025 53 11 . . PUNCT american_scientific_journal-8025 54 1 theorem theorem ADJ american_scientific_journal-8025 54 2 8 8 NUM american_scientific_journal-8025 54 3 : : PUNCT american_scientific_journal-8025 54 4 let let VERB american_scientific_journal-8025 54 5 𝑋 𝑋 NOUN american_scientific_journal-8025 54 6 : : PUNCT american_scientific_journal-8025 54 7 ( ( PUNCT american_scientific_journal-8025 54 8 βˆ’βˆž βˆ’βˆž NOUN american_scientific_journal-8025 54 9 , , PUNCT american_scientific_journal-8025 54 10 ∞ ∞ PROPN american_scientific_journal-8025 54 11 ) ) PUNCT american_scientific_journal-8025 55 1 β†’ β†’ SYM american_scientific_journal-8025 55 2 𝑅 𝑅 NOUN american_scientific_journal-8025 55 3 a a DET american_scientific_journal-8025 55 4 random random ADJ american_scientific_journal-8025 55 5 variable variable NOUN american_scientific_journal-8025 55 6 with with ADP american_scientific_journal-8025 55 7 finite finite ADJ american_scientific_journal-8025 55 8 mean mean NOUN american_scientific_journal-8025 55 9 and and CCONJ american_scientific_journal-8025 55 10 density density NOUN american_scientific_journal-8025 55 11 function function NOUN american_scientific_journal-8025 55 12 given give VERB american_scientific_journal-8025 55 13 by by ADP american_scientific_journal-8025 55 14 Ο†(π‘₯ Ο†(π‘₯ PROPN american_scientific_journal-8025 55 15 ) ) PUNCT american_scientific_journal-8025 55 16 . . PUNCT american_scientific_journal-8025 56 1 suppose suppose VERB american_scientific_journal-8025 56 2 that that SCONJ american_scientific_journal-8025 56 3 exist exist VERB american_scientific_journal-8025 56 4 π‘Ž π‘Ž PRON american_scientific_journal-8025 56 5 > > X american_scientific_journal-8025 56 6 𝐸(𝑋 𝐸(𝑋 NOUN american_scientific_journal-8025 56 7 ) ) PUNCT american_scientific_journal-8025 56 8 such such ADJ american_scientific_journal-8025 56 9 that that SCONJ american_scientific_journal-8025 56 10 1.βˆ€π‘₯ 1.βˆ€π‘₯ NUM american_scientific_journal-8025 56 11 ∈ ∈ NOUN american_scientific_journal-8025 57 1 [ [ X american_scientific_journal-8025 57 2 𝐸[𝑋 𝐸[𝑋 NOUN american_scientific_journal-8025 57 3 ] ] PUNCT american_scientific_journal-8025 57 4 , , PUNCT american_scientific_journal-8025 57 5 π‘Ž π‘Ž NOUN american_scientific_journal-8025 57 6 ) ) PUNCT american_scientific_journal-8025 57 7 , , PUNCT american_scientific_journal-8025 57 8 Ο†(𝐸[𝑋 Ο†(𝐸[𝑋 PROPN american_scientific_journal-8025 57 9 ] ] PUNCT american_scientific_journal-8025 57 10 + + NUM american_scientific_journal-8025 57 11 π‘₯ π‘₯ X american_scientific_journal-8025 57 12 ) ) PUNCT american_scientific_journal-8025 57 13 βˆ’ βˆ’ PROPN american_scientific_journal-8025 57 14 Ο†(𝐸[𝑋 Ο†(𝐸[𝑋 PROPN american_scientific_journal-8025 57 15 ] ] PUNCT american_scientific_journal-8025 57 16 βˆ’ βˆ’ PROPN american_scientific_journal-8025 57 17 π‘₯ π‘₯ NOUN american_scientific_journal-8025 57 18 ) ) PUNCT american_scientific_journal-8025 57 19 ≀ ≀ NOUN american_scientific_journal-8025 57 20 0 0 NUM american_scientific_journal-8025 58 1 2.βˆ€π‘₯ 2.βˆ€π‘₯ NUM american_scientific_journal-8025 58 2 ∈ ∈ PROPN american_scientific_journal-8025 58 3 [ [ X american_scientific_journal-8025 58 4 π‘Ž π‘Ž X american_scientific_journal-8025 58 5 , , PUNCT american_scientific_journal-8025 58 6 ∞ ∞ PROPN american_scientific_journal-8025 58 7 ) ) PUNCT american_scientific_journal-8025 58 8 , , PUNCT american_scientific_journal-8025 58 9 Ο†(𝐸[𝑋 Ο†(𝐸[𝑋 PROPN american_scientific_journal-8025 58 10 ] ] PUNCT american_scientific_journal-8025 59 1 + + NUM american_scientific_journal-8025 59 2 π‘₯ π‘₯ X american_scientific_journal-8025 59 3 ) ) PUNCT american_scientific_journal-8025 59 4 βˆ’ βˆ’ PROPN american_scientific_journal-8025 59 5 Ο†(𝐸[𝑋 Ο†(𝐸[𝑋 PROPN american_scientific_journal-8025 59 6 ] ] PUNCT american_scientific_journal-8025 59 7 βˆ’ βˆ’ PROPN american_scientific_journal-8025 59 8 π‘₯ π‘₯ X american_scientific_journal-8025 59 9 ) ) PUNCT american_scientific_journal-8025 59 10 β‰₯ β‰₯ NOUN american_scientific_journal-8025 59 11 0 0 PUNCT american_scientific_journal-8025 60 1 then then ADV american_scientific_journal-8025 60 2 , , PUNCT american_scientific_journal-8025 60 3 the the DET american_scientific_journal-8025 60 4 random random ADJ american_scientific_journal-8025 60 5 variable variable NOUN american_scientific_journal-8025 60 6 is𝑋 is𝑋 PROPN american_scientific_journal-8025 60 7 is be AUX american_scientific_journal-8025 60 8 strongly strongly ADV american_scientific_journal-8025 60 9 asymmetric asymmetric ADJ american_scientific_journal-8025 60 10 to to ADP american_scientific_journal-8025 60 11 the the DET american_scientific_journal-8025 60 12 right right NOUN american_scientific_journal-8025 60 13 . . PUNCT american_scientific_journal-8025 61 1 proof proof NOUN american_scientific_journal-8025 61 2 . . PUNCT american_scientific_journal-8025 62 1 suppose suppose VERB american_scientific_journal-8025 62 2 without without ADP american_scientific_journal-8025 62 3 loss loss NOUN american_scientific_journal-8025 62 4 of of ADP american_scientific_journal-8025 62 5 generality generality NOUN american_scientific_journal-8025 62 6 that that PRON american_scientific_journal-8025 62 7 𝐸(𝑋 𝐸(𝑋 VERB american_scientific_journal-8025 62 8 ) ) PUNCT american_scientific_journal-8025 62 9 = = SYM american_scientific_journal-8025 62 10 0 0 X american_scientific_journal-8025 62 11 . . PUNCT american_scientific_journal-8025 63 1 let let AUX american_scientific_journal-8025 63 2 be be AUX american_scientific_journal-8025 63 3 𝑔(π‘₯ 𝑔(π‘₯ PRON american_scientific_journal-8025 63 4 ) ) PUNCT american_scientific_journal-8025 64 1 = = SYM american_scientific_journal-8025 64 2 Ο†(π‘₯ Ο†(π‘₯ PROPN american_scientific_journal-8025 64 3 ) ) PUNCT american_scientific_journal-8025 64 4 βˆ’ βˆ’ PROPN american_scientific_journal-8025 64 5 Ο†(βˆ’π‘₯),we Ο†(βˆ’π‘₯),we PROPN american_scientific_journal-8025 64 6 want want VERB american_scientific_journal-8025 64 7 to to PART american_scientific_journal-8025 64 8 prove prove VERB american_scientific_journal-8025 64 9 that that SCONJ american_scientific_journal-8025 64 10 for for ADP american_scientific_journal-8025 64 11 every every DET american_scientific_journal-8025 64 12 odd odd ADJ american_scientific_journal-8025 64 13 - - PUNCT american_scientific_journal-8025 64 14 convex convex NOUN american_scientific_journal-8025 64 15 function function NOUN american_scientific_journal-8025 64 16 𝑓 𝑓 PRON american_scientific_journal-8025 64 17 ( ( PUNCT american_scientific_journal-8025 64 18 11 11 NUM american_scientific_journal-8025 64 19 ) ) PUNCT american_scientific_journal-8025 64 20 due due ADP american_scientific_journal-8025 64 21 to to ADP american_scientific_journal-8025 64 22 the the DET american_scientific_journal-8025 64 23 parity parity NOUN american_scientific_journal-8025 64 24 of of ADP american_scientific_journal-8025 64 25 𝑓 𝑓 PRON american_scientific_journal-8025 64 26 , , PUNCT american_scientific_journal-8025 64 27 , , PUNCT american_scientific_journal-8025 64 28 we we PRON american_scientific_journal-8025 64 29 have have VERB american_scientific_journal-8025 64 30 ( ( PUNCT american_scientific_journal-8025 64 31 12 12 NUM american_scientific_journal-8025 64 32 ) ) PUNCT american_scientific_journal-8025 64 33 since since SCONJ american_scientific_journal-8025 64 34 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 64 35 ) ) PUNCT american_scientific_journal-8025 64 36 is be AUX american_scientific_journal-8025 64 37 convex convex ADJ american_scientific_journal-8025 64 38 in in ADP american_scientific_journal-8025 64 39 𝑅+ 𝑅+ NOUN american_scientific_journal-8025 64 40 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 64 41 ) ) PUNCT american_scientific_journal-8025 64 42 ≀ ≀ NOUN american_scientific_journal-8025 64 43 𝑓(π‘Ž 𝑓(π‘Ž NOUN american_scientific_journal-8025 64 44 ) ) PUNCT american_scientific_journal-8025 64 45 π‘Ž π‘Ž DET american_scientific_journal-8025 64 46 π‘₯ π‘₯ NOUN american_scientific_journal-8025 64 47 , , PUNCT american_scientific_journal-8025 64 48 for for ADP american_scientific_journal-8025 64 49 all all PRON american_scientific_journal-8025 64 50 0 0 NUM american_scientific_journal-8025 64 51 ≀ ≀ NUM american_scientific_journal-8025 64 52 π‘₯ π‘₯ DET american_scientific_journal-8025 64 53 ≀ ≀ NUM american_scientific_journal-8025 64 54 π‘Ž π‘Ž ADP american_scientific_journal-8025 64 55 and and CCONJ american_scientific_journal-8025 64 56 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 64 57 ) ) PUNCT american_scientific_journal-8025 64 58 β‰₯ β‰₯ NOUN american_scientific_journal-8025 64 59 𝑓′(π‘Ž)(π‘₯ 𝑓′(π‘Ž)(π‘₯ ADV american_scientific_journal-8025 64 60 βˆ’ βˆ’ PRON american_scientific_journal-8025 64 61 π‘Ž π‘Ž X american_scientific_journal-8025 64 62 ) ) PUNCT american_scientific_journal-8025 64 63 + + CCONJ american_scientific_journal-8025 64 64 𝑓(π‘Ž 𝑓(π‘Ž ADJ american_scientific_journal-8025 64 65 ) ) PUNCT american_scientific_journal-8025 64 66 , , PUNCT american_scientific_journal-8025 64 67 for for SCONJ american_scientific_journal-8025 64 68 all all DET american_scientific_journal-8025 64 69 π‘₯ π‘₯ PRON american_scientific_journal-8025 64 70 > > X american_scientific_journal-8025 64 71 π‘Ž π‘Ž X american_scientific_journal-8025 64 72 therefore therefore ADV american_scientific_journal-8025 64 73 . . PUNCT american_scientific_journal-8025 65 1 american american ADJ american_scientific_journal-8025 65 2 academic academic ADJ american_scientific_journal-8025 65 3 scientific scientific ADJ american_scientific_journal-8025 65 4 research research NOUN american_scientific_journal-8025 65 5 journal journal NOUN american_scientific_journal-8025 65 6 for for ADP american_scientific_journal-8025 65 7 engineering engineering NOUN american_scientific_journal-8025 65 8 , , PUNCT american_scientific_journal-8025 65 9 technology technology NOUN american_scientific_journal-8025 65 10 , , PUNCT american_scientific_journal-8025 65 11 and and CCONJ american_scientific_journal-8025 65 12 sciences science NOUN american_scientific_journal-8025 65 13 ( ( PUNCT american_scientific_journal-8025 65 14 asrjets asrjet NOUN american_scientific_journal-8025 65 15 ) ) PUNCT american_scientific_journal-8025 65 16 ( ( PUNCT american_scientific_journal-8025 65 17 2022 2022 NUM american_scientific_journal-8025 65 18 ) ) PUNCT american_scientific_journal-8025 65 19 volume volume NOUN american_scientific_journal-8025 65 20 90 90 NUM american_scientific_journal-8025 65 21 , , PUNCT american_scientific_journal-8025 65 22 no1 no1 PROPN american_scientific_journal-8025 65 23 , , PUNCT american_scientific_journal-8025 65 24 pp pp ADV american_scientific_journal-8025 65 25 204 204 NUM american_scientific_journal-8025 65 26 - - SYM american_scientific_journal-8025 65 27 213 213 NUM american_scientific_journal-8025 65 28 208 208 NUM american_scientific_journal-8025 65 29 ( ( PUNCT american_scientific_journal-8025 65 30 13 13 NUM american_scientific_journal-8025 65 31 ) ) PUNCT american_scientific_journal-8025 65 32 we we PRON american_scientific_journal-8025 65 33 conclude conclude VERB american_scientific_journal-8025 65 34 that that PRON american_scientific_journal-8025 65 35 : : PUNCT american_scientific_journal-8025 65 36 ( ( PUNCT american_scientific_journal-8025 65 37 14 14 NUM american_scientific_journal-8025 65 38 ) ) PUNCT american_scientific_journal-8025 65 39 remark remark NOUN american_scientific_journal-8025 65 40 9 9 NUM american_scientific_journal-8025 65 41 : : PUNCT american_scientific_journal-8025 65 42 because because SCONJ american_scientific_journal-8025 65 43 of of ADP american_scientific_journal-8025 65 44 theorem theorem NOUN american_scientific_journal-8025 65 45 8 8 NUM american_scientific_journal-8025 65 46 , , PUNCT american_scientific_journal-8025 65 47 distributions distribution NOUN american_scientific_journal-8025 65 48 such such ADJ american_scientific_journal-8025 65 49 as as ADP american_scientific_journal-8025 65 50 the the DET american_scientific_journal-8025 65 51 gamma gamma PROPN american_scientific_journal-8025 65 52 family family NOUN american_scientific_journal-8025 65 53 are be AUX american_scientific_journal-8025 65 54 strongly strongly ADV american_scientific_journal-8025 65 55 asymmetric asymmetric ADJ american_scientific_journal-8025 65 56 to to ADP american_scientific_journal-8025 65 57 the the DET american_scientific_journal-8025 65 58 right right NOUN american_scientific_journal-8025 65 59 . . PUNCT american_scientific_journal-8025 66 1 theorem theorem ADJ american_scientific_journal-8025 66 2 10 10 NUM american_scientific_journal-8025 66 3 : : PUNCT american_scientific_journal-8025 66 4 a a DET american_scientific_journal-8025 66 5 random random ADJ american_scientific_journal-8025 66 6 variable variable NOUN american_scientific_journal-8025 66 7 𝑋 𝑋 NOUN american_scientific_journal-8025 66 8 is be AUX american_scientific_journal-8025 66 9 strongly strongly ADV american_scientific_journal-8025 66 10 asymmetric asymmetric ADJ american_scientific_journal-8025 66 11 to to ADP american_scientific_journal-8025 66 12 the the DET american_scientific_journal-8025 66 13 right right ADJ american_scientific_journal-8025 66 14 if if SCONJ american_scientific_journal-8025 67 1 and and CCONJ american_scientific_journal-8025 67 2 only only ADV american_scientific_journal-8025 67 3 if if SCONJ american_scientific_journal-8025 67 4 for for ADP american_scientific_journal-8025 67 5 all all PRON american_scientific_journal-8025 67 6 𝑔 𝑔 NOUN american_scientific_journal-8025 67 7 increasing increase VERB american_scientific_journal-8025 67 8 and and CCONJ american_scientific_journal-8025 67 9 odd odd ADV american_scientific_journal-8025 67 10 - - PUNCT american_scientific_journal-8025 67 11 concave concave VERB american_scientific_journal-8025 67 12 , , PUNCT american_scientific_journal-8025 67 13 π‘†π‘˜π‘”(𝑋 π‘†π‘˜π‘”(𝑋 PROPN american_scientific_journal-8025 67 14 ) ) PUNCT american_scientific_journal-8025 67 15 ≀ ≀ NOUN american_scientific_journal-8025 67 16 0 0 NUM american_scientific_journal-8025 67 17 . . PUNCT american_scientific_journal-8025 68 1 proof proof NOUN american_scientific_journal-8025 68 2 . . PUNCT american_scientific_journal-8025 69 1 first first ADV american_scientific_journal-8025 69 2 , , PUNCT american_scientific_journal-8025 69 3 suppose suppose VERB american_scientific_journal-8025 69 4 that that SCONJ american_scientific_journal-8025 69 5 there there PRON american_scientific_journal-8025 69 6 exists exist VERB american_scientific_journal-8025 69 7 a a DET american_scientific_journal-8025 69 8 constant constant ADJ american_scientific_journal-8025 69 9 Ξ· Ξ· NOUN american_scientific_journal-8025 69 10 > > X american_scientific_journal-8025 69 11 0 0 NUM american_scientific_journal-8025 69 12 such such ADJ american_scientific_journal-8025 69 13 that that DET american_scientific_journal-8025 69 14 for for ADP american_scientific_journal-8025 69 15 all all DET american_scientific_journal-8025 69 16 π‘₯ π‘₯ DET american_scientific_journal-8025 69 17 ∈ ∈ PROPN american_scientific_journal-8025 69 18 𝑅+ 𝑅+ NOUN american_scientific_journal-8025 69 19 , , PUNCT american_scientific_journal-8025 69 20 Ξ·π‘₯ Ξ·π‘₯ ADP american_scientific_journal-8025 69 21 β‰₯ β‰₯ NOUN american_scientific_journal-8025 69 22 𝑔(π‘₯)|𝑅+ 𝑔(π‘₯)|𝑅+ NOUN american_scientific_journal-8025 69 23 , , PUNCT american_scientific_journal-8025 69 24 then then ADV american_scientific_journal-8025 69 25 the the DET american_scientific_journal-8025 69 26 function function NOUN american_scientific_journal-8025 69 27 β„Ž(π‘₯ β„Ž(π‘₯ NUM american_scientific_journal-8025 69 28 ) ) PUNCT american_scientific_journal-8025 69 29 = = PUNCT american_scientific_journal-8025 69 30 Ξ·π‘₯ Ξ·π‘₯ ADP american_scientific_journal-8025 69 31 βˆ’ βˆ’ PROPN american_scientific_journal-8025 69 32 𝑔(π‘₯ 𝑔(π‘₯ NUM american_scientific_journal-8025 69 33 ) ) PUNCT american_scientific_journal-8025 69 34 is be AUX american_scientific_journal-8025 69 35 increasing increase VERB american_scientific_journal-8025 69 36 , , PUNCT american_scientific_journal-8025 69 37 odd odd ADJ american_scientific_journal-8025 69 38 and and CCONJ american_scientific_journal-8025 69 39 convex convex ADJ american_scientific_journal-8025 69 40 , , PUNCT american_scientific_journal-8025 69 41 so so ADV american_scientific_journal-8025 69 42 π‘†π‘˜β„Ž(π‘₯ π‘†π‘˜β„Ž(π‘₯ NUM american_scientific_journal-8025 69 43 ) ) PUNCT american_scientific_journal-8025 69 44 β‰₯ β‰₯ NOUN american_scientific_journal-8025 69 45 0 0 NUM american_scientific_journal-8025 69 46 . . PUNCT american_scientific_journal-8025 70 1 now now ADV american_scientific_journal-8025 70 2 suppose suppose VERB american_scientific_journal-8025 70 3 that that SCONJ american_scientific_journal-8025 70 4 there there PRON american_scientific_journal-8025 70 5 is be VERB american_scientific_journal-8025 70 6 no no DET american_scientific_journal-8025 70 7 $ $ SYM american_scientific_journal-8025 70 8 \eta>0 \eta>0 NOUN american_scientific_journal-8025 70 9 $ $ SYM american_scientific_journal-8025 70 10 satisfying satisfy VERB american_scientific_journal-8025 70 11 that that SCONJ american_scientific_journal-8025 70 12 , , PUNCT american_scientific_journal-8025 70 13 for for ADP american_scientific_journal-8025 70 14 all all DET american_scientific_journal-8025 70 15 π‘₯ π‘₯ DET american_scientific_journal-8025 70 16 ∈ ∈ NOUN american_scientific_journal-8025 70 17 𝑅+ 𝑅+ NOUN american_scientific_journal-8025 70 18 , , PUNCT american_scientific_journal-8025 70 19 Ξ·π‘₯ Ξ·π‘₯ VERB american_scientific_journal-8025 70 20 β‰₯ β‰₯ NOUN american_scientific_journal-8025 70 21 𝑔(π‘₯)|𝑅+ 𝑔(π‘₯)|𝑅+ NOUN american_scientific_journal-8025 70 22 , , PUNCT american_scientific_journal-8025 70 23 define define VERB american_scientific_journal-8025 70 24 ( ( PUNCT american_scientific_journal-8025 70 25 15 15 NUM american_scientific_journal-8025 70 26 ) ) PUNCT american_scientific_journal-8025 70 27 where where SCONJ american_scientific_journal-8025 70 28 π‘Ÿπ‘› π‘Ÿπ‘› ADP american_scientific_journal-8025 70 29 > > X american_scientific_journal-8025 70 30 0 0 NUM american_scientific_journal-8025 70 31 satisfies satisfie NOUN american_scientific_journal-8025 70 32 that that DET american_scientific_journal-8025 70 33 π‘›π‘Ÿπ‘› π‘›π‘Ÿπ‘› NOUN american_scientific_journal-8025 70 34 = = SYM american_scientific_journal-8025 70 35 𝑔(π‘Ÿπ‘› 𝑔(π‘Ÿπ‘› NOUN american_scientific_journal-8025 70 36 ) ) PUNCT american_scientific_journal-8025 70 37 , , PUNCT american_scientific_journal-8025 70 38 such such ADJ american_scientific_journal-8025 70 39 π‘Ÿπ‘›exist π‘Ÿπ‘›exist NOUN american_scientific_journal-8025 70 40 because because SCONJ american_scientific_journal-8025 70 41 of of ADP american_scientific_journal-8025 70 42 the the DET american_scientific_journal-8025 70 43 concavity concavity NOUN american_scientific_journal-8025 70 44 of of ADP american_scientific_journal-8025 70 45 𝑔. 𝑔. NOUN american_scientific_journal-8025 70 46 we we PRON american_scientific_journal-8025 70 47 have have VERB american_scientific_journal-8025 70 48 that that PRON american_scientific_journal-8025 70 49 π‘Ÿπ‘› π‘Ÿπ‘› ADP american_scientific_journal-8025 70 50 β†’ β†’ SYM american_scientific_journal-8025 70 51 0 0 NUM american_scientific_journal-8025 70 52 and and CCONJ american_scientific_journal-8025 70 53 that that DET american_scientific_journal-8025 70 54 𝑔𝑛(π‘₯ 𝑔𝑛(π‘₯ NOUN american_scientific_journal-8025 70 55 ) ) PUNCT american_scientific_journal-8025 70 56 converges converge VERB american_scientific_journal-8025 70 57 uniformly uniformly ADV american_scientific_journal-8025 70 58 to to ADP american_scientific_journal-8025 70 59 𝑔(π‘₯ 𝑔(π‘₯ NUM american_scientific_journal-8025 70 60 ) ) PUNCT american_scientific_journal-8025 70 61 . . PUNCT american_scientific_journal-8025 71 1 applying apply VERB american_scientific_journal-8025 71 2 the the DET american_scientific_journal-8025 71 3 first first ADJ american_scientific_journal-8025 71 4 part part NOUN american_scientific_journal-8025 71 5 we we PRON american_scientific_journal-8025 71 6 have have VERB american_scientific_journal-8025 71 7 that that DET american_scientific_journal-8025 71 8 π‘†π‘˜π‘”π‘› π‘†π‘˜π‘”π‘› PROPN american_scientific_journal-8025 71 9 ( ( PUNCT american_scientific_journal-8025 71 10 𝑋 𝑋 PROPN american_scientific_journal-8025 71 11 ) ) PUNCT american_scientific_journal-8025 71 12 ≀ ≀ NOUN american_scientific_journal-8025 71 13 0 0 NUM american_scientific_journal-8025 71 14 and and CCONJ american_scientific_journal-8025 71 15 by by ADP american_scientific_journal-8025 71 16 convergence convergence NOUN american_scientific_journal-8025 71 17 π‘†π‘˜π‘”(𝑋 π‘†π‘˜π‘”(𝑋 PROPN american_scientific_journal-8025 71 18 ) ) PUNCT american_scientific_journal-8025 71 19 ≀ ≀ NOUN american_scientific_journal-8025 71 20 0 0 NUM american_scientific_journal-8025 71 21 . . PUNCT american_scientific_journal-8025 72 1 conversely conversely ADV american_scientific_journal-8025 72 2 consider consider VERB american_scientific_journal-8025 72 3 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 72 4 ) ) PUNCT american_scientific_journal-8025 72 5 odd odd ADJ american_scientific_journal-8025 72 6 concave concave NOUN american_scientific_journal-8025 72 7 and and CCONJ american_scientific_journal-8025 72 8 increasing increase VERB american_scientific_journal-8025 72 9 function function NOUN american_scientific_journal-8025 72 10 . . PUNCT american_scientific_journal-8025 73 1 we we PRON american_scientific_journal-8025 73 2 will will AUX american_scientific_journal-8025 73 3 assume assume VERB american_scientific_journal-8025 73 4 that that SCONJ american_scientific_journal-8025 73 5 𝑓 𝑓 PRON american_scientific_journal-8025 73 6 is be AUX american_scientific_journal-8025 73 7 american american ADJ american_scientific_journal-8025 73 8 academic academic ADJ american_scientific_journal-8025 73 9 scientific scientific ADJ american_scientific_journal-8025 73 10 research research NOUN american_scientific_journal-8025 73 11 journal journal NOUN american_scientific_journal-8025 73 12 for for ADP american_scientific_journal-8025 73 13 engineering engineering NOUN american_scientific_journal-8025 73 14 , , PUNCT american_scientific_journal-8025 73 15 technology technology NOUN american_scientific_journal-8025 73 16 , , PUNCT american_scientific_journal-8025 73 17 and and CCONJ american_scientific_journal-8025 73 18 sciences science NOUN american_scientific_journal-8025 73 19 ( ( PUNCT american_scientific_journal-8025 73 20 asrjets asrjet NOUN american_scientific_journal-8025 73 21 ) ) PUNCT american_scientific_journal-8025 73 22 ( ( PUNCT american_scientific_journal-8025 73 23 2022 2022 NUM american_scientific_journal-8025 73 24 ) ) PUNCT american_scientific_journal-8025 73 25 volume volume NOUN american_scientific_journal-8025 73 26 90 90 NUM american_scientific_journal-8025 73 27 , , PUNCT american_scientific_journal-8025 73 28 no1 no1 PROPN american_scientific_journal-8025 73 29 , , PUNCT american_scientific_journal-8025 73 30 pp pp ADV american_scientific_journal-8025 73 31 204 204 NUM american_scientific_journal-8025 73 32 - - SYM american_scientific_journal-8025 73 33 213 213 NUM american_scientific_journal-8025 73 34 209 209 NUM american_scientific_journal-8025 73 35 differentiable differentiable ADJ american_scientific_journal-8025 73 36 ( ( PUNCT american_scientific_journal-8025 73 37 an an DET american_scientific_journal-8025 73 38 analogous analogous ADJ american_scientific_journal-8025 73 39 analysis analysis NOUN american_scientific_journal-8025 73 40 is be AUX american_scientific_journal-8025 73 41 made make VERB american_scientific_journal-8025 73 42 using use VERB american_scientific_journal-8025 73 43 subgradient subgradient NOUN american_scientific_journal-8025 73 44 ) ) PUNCT american_scientific_journal-8025 73 45 and and CCONJ american_scientific_journal-8025 73 46 that that SCONJ american_scientific_journal-8025 73 47 𝐸(𝑋 𝐸(𝑋 NOUN american_scientific_journal-8025 73 48 ) ) PUNCT american_scientific_journal-8025 73 49 = = SYM american_scientific_journal-8025 74 1 0 0 X american_scientific_journal-8025 74 2 . . X american_scientific_journal-8025 75 1 for for ADP american_scientific_journal-8025 75 2 any any DET american_scientific_journal-8025 75 3 Ξ± Ξ± NOUN american_scientific_journal-8025 75 4 > > X american_scientific_journal-8025 75 5 0 0 PUNCT american_scientific_journal-8025 75 6 consider consider VERB american_scientific_journal-8025 75 7 the the DET american_scientific_journal-8025 75 8 function function NOUN american_scientific_journal-8025 75 9 ( ( PUNCT american_scientific_journal-8025 75 10 16 16 NUM american_scientific_journal-8025 75 11 ) ) PUNCT american_scientific_journal-8025 75 12 then then ADV american_scientific_journal-8025 75 13 𝑔α 𝑔α ADV american_scientific_journal-8025 75 14 is be AUX american_scientific_journal-8025 75 15 oan oan PROPN american_scientific_journal-8025 75 16 dd dd NOUN american_scientific_journal-8025 75 17 - - ADJ american_scientific_journal-8025 75 18 concave concave ADJ american_scientific_journal-8025 75 19 and and CCONJ american_scientific_journal-8025 75 20 increasing increase VERB american_scientific_journal-8025 75 21 function function NOUN american_scientific_journal-8025 75 22 . . PUNCT american_scientific_journal-8025 76 1 by by ADP american_scientific_journal-8025 76 2 hypothesis hypothesis NOUN american_scientific_journal-8025 76 3 , , PUNCT american_scientific_journal-8025 76 4 if if SCONJ american_scientific_journal-8025 76 5 𝐹 𝐹 PROPN american_scientific_journal-8025 76 6 is be AUX american_scientific_journal-8025 76 7 the the DET american_scientific_journal-8025 76 8 distribution distribution NOUN american_scientific_journal-8025 76 9 function function NOUN american_scientific_journal-8025 76 10 of of ADP american_scientific_journal-8025 76 11 the the DET american_scientific_journal-8025 76 12 random random ADJ american_scientific_journal-8025 76 13 variable variable NOUN american_scientific_journal-8025 76 14 𝑋 𝑋 PROPN american_scientific_journal-8025 76 15 , , PUNCT american_scientific_journal-8025 76 16 adding add VERB american_scientific_journal-8025 76 17 ∫ ∫ PROPN american_scientific_journal-8025 76 18 𝑓′(Ξ±)π‘₯𝑑𝐹(π‘₯ 𝑓′(Ξ±)π‘₯𝑑𝐹(π‘₯ VERB american_scientific_journal-8025 76 19 ) ) PUNCT american_scientific_journal-8025 77 1 |π‘₯|>Ξ± |π‘₯|>Ξ± ADP american_scientific_journal-8025 77 2 βˆ’ βˆ’ PROPN american_scientific_journal-8025 77 3 ∫ ∫ PROPN american_scientific_journal-8025 77 4 𝑓′(Ξ±)π‘₯𝑑𝐹(π‘₯ 𝑓′(Ξ±)π‘₯𝑑𝐹(π‘₯ VERB american_scientific_journal-8025 77 5 ) ) PUNCT american_scientific_journal-8025 78 1 |π‘₯|>Ξ± |π‘₯|>Ξ± X american_scientific_journal-8025 79 1 + + NUM american_scientific_journal-8025 79 2 ∫ ∫ PROPN american_scientific_journal-8025 79 3 𝑓(π‘₯)𝑑𝐹(π‘₯ 𝑓(π‘₯)𝑑𝐹(π‘₯ VERB american_scientific_journal-8025 79 4 ) ) PUNCT american_scientific_journal-8025 80 1 |π‘₯|>Ξ± |π‘₯|>Ξ± ADP american_scientific_journal-8025 80 2 βˆ’ βˆ’ PROPN american_scientific_journal-8025 80 3 ∫ ∫ PROPN american_scientific_journal-8025 80 4 𝑓(π‘₯)𝑑𝐹(π‘₯ 𝑓(π‘₯)𝑑𝐹(π‘₯ VERB american_scientific_journal-8025 80 5 ) ) PUNCT american_scientific_journal-8025 80 6 |π‘₯|>Ξ± |π‘₯|>Ξ± NOUN american_scientific_journal-8025 80 7 , , PUNCT american_scientific_journal-8025 80 8 and and CCONJ american_scientific_journal-8025 80 9 organizing organize VERB american_scientific_journal-8025 80 10 the the DET american_scientific_journal-8025 80 11 terms term NOUN american_scientific_journal-8025 80 12 we we PRON american_scientific_journal-8025 80 13 obtain obtain VERB american_scientific_journal-8025 80 14 that that PRON american_scientific_journal-8025 80 15 for for ADP american_scientific_journal-8025 80 16 all all DET american_scientific_journal-8025 80 17 Ξ± Ξ± NOUN american_scientific_journal-8025 80 18 > > X american_scientific_journal-8025 80 19 0 0 NUM american_scientific_journal-8025 80 20 , , PUNCT american_scientific_journal-8025 80 21 and and CCONJ american_scientific_journal-8025 80 22 finally finally ADV american_scientific_journal-8025 80 23 , , PUNCT american_scientific_journal-8025 80 24 we we PRON american_scientific_journal-8025 80 25 have have VERB american_scientific_journal-8025 80 26 that that PRON american_scientific_journal-8025 80 27 . . PUNCT american_scientific_journal-8025 81 1 so so ADV american_scientific_journal-8025 81 2 ∫ ∫ PROPN american_scientific_journal-8025 81 3 𝑓(π‘₯)𝑑𝐹(π‘₯ 𝑓(π‘₯)𝑑𝐹(π‘₯ VERB american_scientific_journal-8025 81 4 ) ) PUNCT american_scientific_journal-8025 82 1 β‰₯ β‰₯ NOUN american_scientific_journal-8025 82 2 0 0 NUM american_scientific_journal-8025 82 3 2.1 2.1 NUM american_scientific_journal-8025 82 4 . . PUNCT american_scientific_journal-8025 83 1 strong strong ADJ american_scientific_journal-8025 83 2 asymmetry asymmetry NOUN american_scientific_journal-8025 83 3 a a DET american_scientific_journal-8025 83 4 common common ADJ american_scientific_journal-8025 83 5 conceptual conceptual ADJ american_scientific_journal-8025 83 6 mistake mistake NOUN american_scientific_journal-8025 83 7 is be AUX american_scientific_journal-8025 83 8 to to PART american_scientific_journal-8025 83 9 say say VERB american_scientific_journal-8025 83 10 that that SCONJ american_scientific_journal-8025 83 11 if if SCONJ american_scientific_journal-8025 83 12 a a DET american_scientific_journal-8025 83 13 distribution distribution NOUN american_scientific_journal-8025 83 14 $ $ SYM american_scientific_journal-8025 83 15 f$ f$ NOUN american_scientific_journal-8025 83 16 is be AUX american_scientific_journal-8025 83 17 unimodal unimodal ADJ american_scientific_journal-8025 83 18 and and CCONJ american_scientific_journal-8025 83 19 the the DET american_scientific_journal-8025 83 20 pearson pearson PROPN american_scientific_journal-8025 83 21 skewness skewness PROPN american_scientific_journal-8025 83 22 coefficient coefficient NOUN american_scientific_journal-8025 83 23 is be AUX american_scientific_journal-8025 83 24 positive positive ADJ american_scientific_journal-8025 83 25 , , PUNCT american_scientific_journal-8025 83 26 then then ADV american_scientific_journal-8025 83 27 it it PRON american_scientific_journal-8025 83 28 is be AUX american_scientific_journal-8025 83 29 true true ADJ american_scientific_journal-8025 83 30 that that SCONJ american_scientific_journal-8025 83 31 π‘šπ‘œπ‘‘π‘’(𝐹 π‘šπ‘œπ‘‘π‘’(𝐹 NOUN american_scientific_journal-8025 83 32 ) ) PUNCT american_scientific_journal-8025 83 33 ≀ ≀ NOUN american_scientific_journal-8025 83 34 π‘šπ‘’π‘‘π‘–π‘Žπ‘›(𝐹 π‘šπ‘’π‘‘π‘–π‘Žπ‘›(𝐹 NUM american_scientific_journal-8025 83 35 ) ) PUNCT american_scientific_journal-8025 83 36 ≀ ≀ NUM american_scientific_journal-8025 83 37 π‘šπ‘’π‘Žπ‘›(𝐹 π‘šπ‘’π‘Žπ‘›(𝐹 NOUN american_scientific_journal-8025 83 38 ) ) PUNCT american_scientific_journal-8025 83 39 . . PUNCT american_scientific_journal-8025 84 1 \cite{abadir2005mean \cite{abadir2005mean PROPN american_scientific_journal-8025 84 2 } } PUNCT american_scientific_journal-8025 84 3 showed show VERB american_scientific_journal-8025 84 4 several several ADJ american_scientific_journal-8025 84 5 examples example NOUN american_scientific_journal-8025 84 6 of of ADP american_scientific_journal-8025 84 7 the the DET american_scientific_journal-8025 84 8 violations violation NOUN american_scientific_journal-8025 84 9 of of ADP american_scientific_journal-8025 84 10 each each DET american_scientific_journal-8025 84 11 one one NUM american_scientific_journal-8025 84 12 of of ADP american_scientific_journal-8025 84 13 the the DET american_scientific_journal-8025 84 14 inequalities inequality NOUN american_scientific_journal-8025 84 15 . . PUNCT american_scientific_journal-8025 85 1 when when SCONJ american_scientific_journal-8025 85 2 a a DET american_scientific_journal-8025 85 3 strongly strongly ADV american_scientific_journal-8025 85 4 asymmetric asymmetric ADJ american_scientific_journal-8025 85 5 is be AUX american_scientific_journal-8025 85 6 considered consider VERB american_scientific_journal-8025 85 7 , , PUNCT american_scientific_journal-8025 85 8 it it PRON american_scientific_journal-8025 85 9 is be AUX american_scientific_journal-8025 85 10 possible possible ADJ american_scientific_journal-8025 85 11 to to PART american_scientific_journal-8025 85 12 show show VERB american_scientific_journal-8025 85 13 that that PRON american_scientific_journal-8025 85 14 π‘šπ‘’π‘‘π‘–π‘Žπ‘›(𝐹 π‘šπ‘’π‘‘π‘–π‘Žπ‘›(𝐹 NOUN american_scientific_journal-8025 85 15 ) ) PUNCT american_scientific_journal-8025 85 16 ≀ ≀ NUM american_scientific_journal-8025 85 17 π‘šπ‘’π‘Žπ‘›(𝐹 π‘šπ‘’π‘Žπ‘›(𝐹 NOUN american_scientific_journal-8025 85 18 ) ) PUNCT american_scientific_journal-8025 85 19 theorem theorem VERB american_scientific_journal-8025 85 20 11 11 NUM american_scientific_journal-8025 85 21 . . PUNCT american_scientific_journal-8025 86 1 if if SCONJ american_scientific_journal-8025 86 2 𝑋 𝑋 PROPN american_scientific_journal-8025 86 3 is be AUX american_scientific_journal-8025 86 4 a a DET american_scientific_journal-8025 86 5 random random ADJ american_scientific_journal-8025 86 6 variable variable NOUN american_scientific_journal-8025 86 7 strongly strongly ADV american_scientific_journal-8025 86 8 asymmetric asymmetric ADJ american_scientific_journal-8025 86 9 to to ADP american_scientific_journal-8025 86 10 the the DET american_scientific_journal-8025 86 11 right right NOUN american_scientific_journal-8025 86 12 , , PUNCT american_scientific_journal-8025 86 13 then then ADV american_scientific_journal-8025 86 14 the the DET american_scientific_journal-8025 86 15 median median NOUN american_scientific_journal-8025 86 16 of of ADP american_scientific_journal-8025 86 17 𝑋 𝑋 PROPN american_scientific_journal-8025 86 18 is be AUX american_scientific_journal-8025 86 19 less less ADV american_scientific_journal-8025 86 20 or or CCONJ american_scientific_journal-8025 86 21 equal equal ADJ american_scientific_journal-8025 86 22 to to ADP american_scientific_journal-8025 86 23 the the DET american_scientific_journal-8025 86 24 mean mean NOUN american_scientific_journal-8025 86 25 of of ADP american_scientific_journal-8025 86 26 𝑋. 𝑋. PROPN american_scientific_journal-8025 86 27 proof proof NOUN american_scientific_journal-8025 86 28 . . PUNCT american_scientific_journal-8025 87 1 let let VERB american_scientific_journal-8025 87 2 β„Ž(π‘₯ β„Ž(π‘₯ NUM american_scientific_journal-8025 87 3 ) ) PUNCT american_scientific_journal-8025 87 4 = = PUNCT american_scientific_journal-8025 88 1 1π‘₯>0 1π‘₯>0 NUM american_scientific_journal-8025 89 1 βˆ’ βˆ’ NUM american_scientific_journal-8025 89 2 1π‘₯<0 1π‘₯<0 NUM american_scientific_journal-8025 89 3 and and CCONJ american_scientific_journal-8025 89 4 𝑔𝑛(π‘₯ 𝑔𝑛(π‘₯ NOUN american_scientific_journal-8025 89 5 ) ) PUNCT american_scientific_journal-8025 89 6 a a DET american_scientific_journal-8025 89 7 sequence sequence NOUN american_scientific_journal-8025 89 8 of of ADP american_scientific_journal-8025 89 9 functions function NOUN american_scientific_journal-8025 89 10 converging converge VERB american_scientific_journal-8025 89 11 to to ADP american_scientific_journal-8025 89 12 β„Ž(π‘₯ β„Ž(π‘₯ NOUN american_scientific_journal-8025 89 13 ) ) PUNCT american_scientific_journal-8025 89 14 such such ADJ american_scientific_journal-8025 89 15 that that SCONJ american_scientific_journal-8025 89 16 each each DET american_scientific_journal-8025 89 17 𝑔𝑛(π‘₯ 𝑔𝑛(π‘₯ NOUN american_scientific_journal-8025 89 18 ) ) PUNCT american_scientific_journal-8025 89 19 is be AUX american_scientific_journal-8025 89 20 odd odd ADJ american_scientific_journal-8025 89 21 concave concave NOUN american_scientific_journal-8025 89 22 and and CCONJ american_scientific_journal-8025 89 23 increasing increase VERB american_scientific_journal-8025 89 24 ( ( PUNCT american_scientific_journal-8025 89 25 note note VERB american_scientific_journal-8025 89 26 that that SCONJ american_scientific_journal-8025 89 27 there there PRON american_scientific_journal-8025 89 28 is be VERB american_scientific_journal-8025 89 29 no no DET american_scientific_journal-8025 89 30 sequence sequence NOUN american_scientific_journal-8025 89 31 of of ADP american_scientific_journal-8025 89 32 odd odd ADJ american_scientific_journal-8025 89 33 - - PUNCT american_scientific_journal-8025 89 34 convex convex NOUN american_scientific_journal-8025 89 35 functions function NOUN american_scientific_journal-8025 89 36 converging converge VERB american_scientific_journal-8025 89 37 to to ADP american_scientific_journal-8025 89 38 β„Ž(π‘₯ β„Ž(π‘₯ NUM american_scientific_journal-8025 89 39 ) ) PUNCT american_scientific_journal-8025 89 40 ) ) PUNCT american_scientific_journal-8025 89 41 , , PUNCT american_scientific_journal-8025 89 42 then then ADV american_scientific_journal-8025 89 43 π‘†π‘˜π‘”π‘› π‘†π‘˜π‘”π‘› PROPN american_scientific_journal-8025 89 44 ( ( PUNCT american_scientific_journal-8025 89 45 π‘₯ π‘₯ NOUN american_scientific_journal-8025 89 46 ) ) PUNCT american_scientific_journal-8025 89 47 ≀ ≀ NOUN american_scientific_journal-8025 89 48 0 0 NUM american_scientific_journal-8025 89 49 . . PUNCT american_scientific_journal-8025 90 1 let let VERB american_scientific_journal-8025 90 2 𝐹 𝐹 PRON american_scientific_journal-8025 90 3 the the DET american_scientific_journal-8025 90 4 distribution distribution NOUN american_scientific_journal-8025 90 5 function function NOUN american_scientific_journal-8025 90 6 of of ADP american_scientific_journal-8025 90 7 the the DET american_scientific_journal-8025 90 8 random random ADJ american_scientific_journal-8025 90 9 variable variable NOUN american_scientific_journal-8025 90 10 𝑋 𝑋 PROPN american_scientific_journal-8025 90 11 then then ADV american_scientific_journal-8025 90 12 , , PUNCT american_scientific_journal-8025 90 13 american american ADJ american_scientific_journal-8025 90 14 academic academic ADJ american_scientific_journal-8025 90 15 scientific scientific ADJ american_scientific_journal-8025 90 16 research research NOUN american_scientific_journal-8025 90 17 journal journal NOUN american_scientific_journal-8025 90 18 for for ADP american_scientific_journal-8025 90 19 engineering engineering NOUN american_scientific_journal-8025 90 20 , , PUNCT american_scientific_journal-8025 90 21 technology technology NOUN american_scientific_journal-8025 90 22 , , PUNCT american_scientific_journal-8025 90 23 and and CCONJ american_scientific_journal-8025 90 24 sciences science NOUN american_scientific_journal-8025 90 25 ( ( PUNCT american_scientific_journal-8025 90 26 asrjets asrjet NOUN american_scientific_journal-8025 90 27 ) ) PUNCT american_scientific_journal-8025 90 28 ( ( PUNCT american_scientific_journal-8025 90 29 2022 2022 NUM american_scientific_journal-8025 90 30 ) ) PUNCT american_scientific_journal-8025 90 31 volume volume NOUN american_scientific_journal-8025 90 32 90 90 NUM american_scientific_journal-8025 90 33 , , PUNCT american_scientific_journal-8025 90 34 no1 no1 PROPN american_scientific_journal-8025 90 35 , , PUNCT american_scientific_journal-8025 90 36 pp pp ADV american_scientific_journal-8025 90 37 204 204 NUM american_scientific_journal-8025 90 38 - - SYM american_scientific_journal-8025 90 39 213 213 NUM american_scientific_journal-8025 90 40 210 210 NUM american_scientific_journal-8025 90 41 ( ( PUNCT american_scientific_journal-8025 90 42 17 17 NUM american_scientific_journal-8025 90 43 ) ) PUNCT american_scientific_journal-8025 90 44 because because SCONJ american_scientific_journal-8025 90 45 |𝑔𝑛(π‘₯)| |𝑔𝑛(π‘₯)| NOUN american_scientific_journal-8025 90 46 ≀ ≀ NOUN american_scientific_journal-8025 90 47 |β„Ž(π‘₯)| |β„Ž(π‘₯)| PROPN american_scientific_journal-8025 90 48 = = SYM american_scientific_journal-8025 90 49 1 1 NUM american_scientific_journal-8025 90 50 , , PUNCT american_scientific_journal-8025 90 51 we we PRON american_scientific_journal-8025 90 52 can can AUX american_scientific_journal-8025 90 53 apply apply VERB american_scientific_journal-8025 90 54 the the DET american_scientific_journal-8025 90 55 dominated dominate VERB american_scientific_journal-8025 90 56 convergence convergence NOUN american_scientific_journal-8025 90 57 theorem theorem VERB american_scientific_journal-8025 90 58 . . PUNCT american_scientific_journal-8025 91 1 finally finally ADV american_scientific_journal-8025 91 2 , , PUNCT american_scientific_journal-8025 91 3 we we PRON american_scientific_journal-8025 91 4 conclude conclude VERB american_scientific_journal-8025 91 5 that that SCONJ american_scientific_journal-8025 91 6 the the DET american_scientific_journal-8025 91 7 median median NOUN american_scientific_journal-8025 91 8 must must AUX american_scientific_journal-8025 91 9 be be AUX american_scientific_journal-8025 91 10 lesser less ADJ american_scientific_journal-8025 91 11 or or CCONJ american_scientific_journal-8025 91 12 equal equal ADJ american_scientific_journal-8025 91 13 to to ADP american_scientific_journal-8025 91 14 𝐸[𝑋 𝐸[𝑋 NOUN american_scientific_journal-8025 91 15 ] ] PUNCT american_scientific_journal-8025 91 16 . . PUNCT american_scientific_journal-8025 92 1 theorem theorem NOUN american_scientific_journal-8025 92 2 12 12 NUM american_scientific_journal-8025 92 3 . . PUNCT american_scientific_journal-8025 93 1 if if SCONJ american_scientific_journal-8025 93 2 𝑋 𝑋 PROPN american_scientific_journal-8025 93 3 is be AUX american_scientific_journal-8025 93 4 a a DET american_scientific_journal-8025 93 5 uni uni ADJ american_scientific_journal-8025 93 6 - - ADJ american_scientific_journal-8025 93 7 modal modal ADJ american_scientific_journal-8025 93 8 random random ADJ american_scientific_journal-8025 93 9 variable variable NOUN american_scientific_journal-8025 93 10 strongly strongly ADV american_scientific_journal-8025 93 11 asymmetric asymmetric ADJ american_scientific_journal-8025 93 12 to to ADP american_scientific_journal-8025 93 13 the the DET american_scientific_journal-8025 93 14 right right NOUN american_scientific_journal-8025 93 15 , , PUNCT american_scientific_journal-8025 93 16 then then ADV american_scientific_journal-8025 93 17 the the DET american_scientific_journal-8025 93 18 mode mode NOUN american_scientific_journal-8025 93 19 of of ADP american_scientific_journal-8025 93 20 𝑋 𝑋 PROPN american_scientific_journal-8025 93 21 is be AUX american_scientific_journal-8025 93 22 less less ADV american_scientific_journal-8025 93 23 or or CCONJ american_scientific_journal-8025 93 24 equal equal ADJ american_scientific_journal-8025 93 25 to to ADP american_scientific_journal-8025 93 26 its its PRON american_scientific_journal-8025 93 27 mean mean NOUN american_scientific_journal-8025 93 28 . . PUNCT american_scientific_journal-8025 94 1 proof proof NOUN american_scientific_journal-8025 94 2 . . PUNCT american_scientific_journal-8025 95 1 we we PRON american_scientific_journal-8025 95 2 will will AUX american_scientific_journal-8025 95 3 proceed proceed VERB american_scientific_journal-8025 95 4 by by ADP american_scientific_journal-8025 95 5 contradiction contradiction NOUN american_scientific_journal-8025 95 6 supposing suppose VERB american_scientific_journal-8025 95 7 that that DET american_scientific_journal-8025 95 8 𝐸[𝑋 𝐸[𝑋 NOUN american_scientific_journal-8025 95 9 ] ] PUNCT american_scientific_journal-8025 96 1 = = SYM american_scientific_journal-8025 96 2 0 0 NUM american_scientific_journal-8025 96 3 and and CCONJ american_scientific_journal-8025 96 4 π‘€π‘œπ‘‘π‘’(𝑋 π‘€π‘œπ‘‘π‘’(𝑋 ADJ american_scientific_journal-8025 96 5 ) ) PUNCT american_scientific_journal-8025 96 6 = = PUNCT american_scientific_journal-8025 97 1 π‘Ž π‘Ž X american_scientific_journal-8025 97 2 > > X american_scientific_journal-8025 97 3 0 0 X american_scientific_journal-8025 97 4 . . PUNCT american_scientific_journal-8025 98 1 let let AUX american_scientific_journal-8025 98 2 be be AUX american_scientific_journal-8025 98 3 β„Žπ‘₯(π‘₯ β„Žπ‘₯(π‘₯ VERB american_scientific_journal-8025 98 4 ) ) PUNCT american_scientific_journal-8025 98 5 the the DET american_scientific_journal-8025 98 6 correspondent correspondent NOUN american_scientific_journal-8025 98 7 density density NOUN american_scientific_journal-8025 98 8 function function NOUN american_scientific_journal-8025 98 9 and and CCONJ american_scientific_journal-8025 98 10 define define VERB american_scientific_journal-8025 98 11 𝑔(π‘₯ 𝑔(π‘₯ NUM american_scientific_journal-8025 98 12 ) ) PUNCT american_scientific_journal-8025 98 13 = = SYM american_scientific_journal-8025 99 1 β„Žπ‘₯(π‘₯ β„Žπ‘₯(π‘₯ ADJ american_scientific_journal-8025 99 2 ) ) PUNCT american_scientific_journal-8025 100 1 βˆ’ βˆ’ PROPN american_scientific_journal-8025 100 2 β„Žπ‘₯(βˆ’π‘₯ β„Žπ‘₯(βˆ’π‘₯ PROPN american_scientific_journal-8025 100 3 ) ) PUNCT american_scientific_journal-8025 100 4 . . PUNCT american_scientific_journal-8025 101 1 we we PRON american_scientific_journal-8025 101 2 have have VERB american_scientific_journal-8025 101 3 that that SCONJ american_scientific_journal-8025 101 4 β„Ž(π‘₯ β„Ž(π‘₯ NUM american_scientific_journal-8025 101 5 ) ) PUNCT american_scientific_journal-8025 101 6 is be AUX american_scientific_journal-8025 101 7 increasing increase VERB american_scientific_journal-8025 101 8 close close ADV american_scientific_journal-8025 101 9 to to ADP american_scientific_journal-8025 101 10 0 0 NUM american_scientific_journal-8025 101 11 , , PUNCT american_scientific_journal-8025 101 12 then then ADV american_scientific_journal-8025 101 13 there there PRON american_scientific_journal-8025 101 14 exist exist VERB american_scientific_journal-8025 101 15 Ξ΅ Ξ΅ PROPN american_scientific_journal-8025 101 16 > > X american_scientific_journal-8025 101 17 0 0 NUM american_scientific_journal-8025 101 18 such such ADJ american_scientific_journal-8025 101 19 that that SCONJ american_scientific_journal-8025 101 20 𝑔(π‘₯ 𝑔(π‘₯ PROPN american_scientific_journal-8025 101 21 ) ) PUNCT american_scientific_journal-8025 101 22 > > X american_scientific_journal-8025 101 23 0 0 PUNCT american_scientific_journal-8025 101 24 for for ADP american_scientific_journal-8025 101 25 all all PRON american_scientific_journal-8025 101 26 π‘₯ π‘₯ DET american_scientific_journal-8025 101 27 ∈ ∈ NOUN american_scientific_journal-8025 101 28 ( ( PUNCT american_scientific_journal-8025 101 29 0 0 NUM american_scientific_journal-8025 101 30 , , PUNCT american_scientific_journal-8025 101 31 Ξ΅ Ξ΅ PROPN american_scientific_journal-8025 101 32 ] ] PUNCT american_scientific_journal-8025 101 33 . . PUNCT american_scientific_journal-8025 102 1 because because SCONJ american_scientific_journal-8025 102 2 ∫ ∫ PROPN american_scientific_journal-8025 102 3 π‘₯𝑔(π‘₯)𝑑π‘₯ π‘₯𝑔(π‘₯)𝑑π‘₯ PROPN american_scientific_journal-8025 102 4 ∞ ∞ PROPN american_scientific_journal-8025 102 5 0 0 NUM american_scientific_journal-8025 103 1 = = SYM american_scientific_journal-8025 103 2 0 0 NUM american_scientific_journal-8025 103 3 there there PRON american_scientific_journal-8025 103 4 exist exist VERB american_scientific_journal-8025 103 5 the the DET american_scientific_journal-8025 103 6 minimum minimum ADJ american_scientific_journal-8025 103 7 positive positive ADJ american_scientific_journal-8025 103 8 root root NOUN american_scientific_journal-8025 103 9 of of ADP american_scientific_journal-8025 103 10 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-8025 103 11 β‰  β‰  PROPN american_scientific_journal-8025 103 12 0 0 NUM american_scientific_journal-8025 103 13 of of ADP american_scientific_journal-8025 103 14 𝑔(π‘₯ 𝑔(π‘₯ PROPN american_scientific_journal-8025 103 15 ) ) PUNCT american_scientific_journal-8025 103 16 . . PUNCT american_scientific_journal-8025 104 1 let let VERB american_scientific_journal-8025 104 2 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 104 3 ) ) PUNCT american_scientific_journal-8025 104 4 = = SYM american_scientific_journal-8025 104 5 ( ( PUNCT american_scientific_journal-8025 105 1 π‘₯ π‘₯ X american_scientific_journal-8025 105 2 βˆ’ βˆ’ X american_scientific_journal-8025 105 3 π‘Ÿ)1[π‘Ÿ,∞ π‘Ÿ)1[π‘Ÿ,∞ NOUN american_scientific_journal-8025 105 4 ) ) PUNCT american_scientific_journal-8025 105 5 , , PUNCT american_scientific_journal-8025 105 6 clearly clearly ADV american_scientific_journal-8025 105 7 f f PROPN american_scientific_journal-8025 105 8 is be AUX american_scientific_journal-8025 105 9 non non NOUN american_scientific_journal-8025 105 10 decreasing decrease VERB american_scientific_journal-8025 105 11 , , PUNCT american_scientific_journal-8025 105 12 positive positive ADJ american_scientific_journal-8025 105 13 and and CCONJ american_scientific_journal-8025 105 14 convex convex ADJ american_scientific_journal-8025 105 15 . . PUNCT american_scientific_journal-8025 106 1 ( ( PUNCT american_scientific_journal-8025 106 2 18 18 NUM american_scientific_journal-8025 106 3 ) ) PUNCT american_scientific_journal-8025 106 4 which which PRON american_scientific_journal-8025 106 5 contradicts contradict VERB american_scientific_journal-8025 106 6 the the DET american_scientific_journal-8025 106 7 hypothesis hypothesis NOUN american_scientific_journal-8025 106 8 of of ADP american_scientific_journal-8025 106 9 strong strong ADJ american_scientific_journal-8025 106 10 asymmetry asymmetry NOUN american_scientific_journal-8025 106 11 to to ADP american_scientific_journal-8025 106 12 the the DET american_scientific_journal-8025 106 13 right right NOUN american_scientific_journal-8025 106 14 . . PUNCT american_scientific_journal-8025 107 1 remark remark PROPN american_scientific_journal-8025 107 2 13 13 NUM american_scientific_journal-8025 107 3 . . PUNCT american_scientific_journal-8025 108 1 even even ADV american_scientific_journal-8025 108 2 in in ADP american_scientific_journal-8025 108 3 the the DET american_scientific_journal-8025 108 4 case case NOUN american_scientific_journal-8025 108 5 of of ADP american_scientific_journal-8025 108 6 strong strong ADJ american_scientific_journal-8025 108 7 asymmetry asymmetry NOUN american_scientific_journal-8025 108 8 , , PUNCT american_scientific_journal-8025 108 9 there there PRON american_scientific_journal-8025 108 10 is be VERB american_scientific_journal-8025 108 11 no no DET american_scientific_journal-8025 108 12 relation relation NOUN american_scientific_journal-8025 108 13 between between ADP american_scientific_journal-8025 108 14 mode mode NOUN american_scientific_journal-8025 108 15 and and CCONJ american_scientific_journal-8025 108 16 median median NOUN american_scientific_journal-8025 108 17 . . PUNCT american_scientific_journal-8025 109 1 in in ADP american_scientific_journal-8025 109 2 fact fact NOUN american_scientific_journal-8025 109 3 for for ADP american_scientific_journal-8025 109 4 ( ( PUNCT american_scientific_journal-8025 109 5 see see VERB american_scientific_journal-8025 109 6 figure figure NOUN american_scientific_journal-8025 109 7 1 1 NUM american_scientific_journal-8025 109 8 ) ) PUNCT american_scientific_journal-8025 109 9 ( ( PUNCT american_scientific_journal-8025 109 10 19 19 NUM american_scientific_journal-8025 109 11 ) ) PUNCT american_scientific_journal-8025 109 12 we we PRON american_scientific_journal-8025 109 13 have have VERB american_scientific_journal-8025 109 14 that that SCONJ american_scientific_journal-8025 109 15 the the DET american_scientific_journal-8025 109 16 distribution distribution NOUN american_scientific_journal-8025 109 17 satisfies satisfy VERB american_scientific_journal-8025 109 18 the the DET american_scientific_journal-8025 109 19 criteria criterion NOUN american_scientific_journal-8025 109 20 of of ADP american_scientific_journal-8025 109 21 theorem theorem NOUN american_scientific_journal-8025 109 22 8 8 NUM american_scientific_journal-8025 109 23 for for ADP american_scientific_journal-8025 109 24 strong strong ADJ american_scientific_journal-8025 109 25 asymmetry asymmetry NOUN american_scientific_journal-8025 109 26 and and CCONJ american_scientific_journal-8025 109 27 π‘šπ‘’π‘‘π‘–π‘Žπ‘› π‘šπ‘’π‘‘π‘–π‘Žπ‘› NOUN american_scientific_journal-8025 109 28 = = SYM american_scientific_journal-8025 109 29 βˆ’1.0167 βˆ’1.0167 ADJ american_scientific_journal-8025 109 30 < < X american_scientific_journal-8025 109 31 π‘šπ‘œπ‘‘π‘’ π‘šπ‘œπ‘‘π‘’ X american_scientific_journal-8025 109 32 = = SYM american_scientific_journal-8025 109 33 βˆ’1 βˆ’1 NOUN american_scientific_journal-8025 109 34 < < X american_scientific_journal-8025 109 35 π‘šπ‘’π‘Žπ‘› π‘šπ‘’π‘Žπ‘› NOUN american_scientific_journal-8025 109 36 = = SYM american_scientific_journal-8025 109 37 βˆ’0.9983 βˆ’0.9983 NOUN american_scientific_journal-8025 109 38 american american ADJ american_scientific_journal-8025 109 39 academic academic ADJ american_scientific_journal-8025 109 40 scientific scientific ADJ american_scientific_journal-8025 109 41 research research NOUN american_scientific_journal-8025 109 42 journal journal NOUN american_scientific_journal-8025 109 43 for for ADP american_scientific_journal-8025 109 44 engineering engineering NOUN american_scientific_journal-8025 109 45 , , PUNCT american_scientific_journal-8025 109 46 technology technology NOUN american_scientific_journal-8025 109 47 , , PUNCT american_scientific_journal-8025 109 48 and and CCONJ american_scientific_journal-8025 109 49 sciences science NOUN american_scientific_journal-8025 109 50 ( ( PUNCT american_scientific_journal-8025 109 51 asrjets asrjet NOUN american_scientific_journal-8025 109 52 ) ) PUNCT american_scientific_journal-8025 109 53 ( ( PUNCT american_scientific_journal-8025 109 54 2022 2022 NUM american_scientific_journal-8025 109 55 ) ) PUNCT american_scientific_journal-8025 109 56 volume volume NOUN american_scientific_journal-8025 109 57 90 90 NUM american_scientific_journal-8025 109 58 , , PUNCT american_scientific_journal-8025 109 59 no1 no1 PROPN american_scientific_journal-8025 109 60 , , PUNCT american_scientific_journal-8025 109 61 pp pp ADV american_scientific_journal-8025 109 62 204 204 NUM american_scientific_journal-8025 109 63 - - SYM american_scientific_journal-8025 109 64 213 213 NUM american_scientific_journal-8025 109 65 211 211 NUM american_scientific_journal-8025 109 66 figure figure NOUN american_scientific_journal-8025 109 67 1 1 NUM american_scientific_journal-8025 109 68 : : PUNCT american_scientific_journal-8025 109 69 strong strong ADJ american_scientific_journal-8025 109 70 distribution distribution NOUN american_scientific_journal-8025 109 71 to to ADP american_scientific_journal-8025 109 72 the the DET american_scientific_journal-8025 109 73 right right NOUN american_scientific_journal-8025 109 74 with with ADP american_scientific_journal-8025 109 75 median median PROPN american_scientific_journal-8025 109 76 < < X american_scientific_journal-8025 109 77 mode mode NOUN american_scientific_journal-8025 109 78 < < X american_scientific_journal-8025 109 79 mean mean NOUN american_scientific_journal-8025 109 80 . . PUNCT american_scientific_journal-8025 110 1 in in ADP american_scientific_journal-8025 110 2 [ [ X american_scientific_journal-8025 110 3 11 11 NUM american_scientific_journal-8025 110 4 ] ] PUNCT american_scientific_journal-8025 110 5 is be AUX american_scientific_journal-8025 110 6 proved prove VERB american_scientific_journal-8025 110 7 that that SCONJ american_scientific_journal-8025 110 8 the the DET american_scientific_journal-8025 110 9 class class NOUN american_scientific_journal-8025 110 10 of of ADP american_scientific_journal-8025 110 11 distributions distribution NOUN american_scientific_journal-8025 110 12 for for ADP american_scientific_journal-8025 110 13 which which PRON american_scientific_journal-8025 110 14 the the DET american_scientific_journal-8025 110 15 median median NOUN american_scientific_journal-8025 110 16 is be AUX american_scientific_journal-8025 110 17 located locate VERB american_scientific_journal-8025 110 18 between between ADP american_scientific_journal-8025 110 19 the the DET american_scientific_journal-8025 110 20 mode mode NOUN american_scientific_journal-8025 110 21 and and CCONJ american_scientific_journal-8025 110 22 the the DET american_scientific_journal-8025 110 23 mean mean NOUN american_scientific_journal-8025 110 24 is be AUX american_scientific_journal-8025 110 25 characterized characterize VERB american_scientific_journal-8025 110 26 by by ADP american_scientific_journal-8025 110 27 the the DET american_scientific_journal-8025 110 28 relation relation NOUN american_scientific_journal-8025 110 29 βˆ€π‘’ βˆ€π‘’ X american_scientific_journal-8025 110 30 > > X american_scientific_journal-8025 110 31 0 0 NUM american_scientific_journal-8025 110 32 , , PUNCT american_scientific_journal-8025 110 33 𝐹(π‘š 𝐹(π‘š ADP american_scientific_journal-8025 110 34 βˆ’ βˆ’ PROPN american_scientific_journal-8025 110 35 𝑒 𝑒 X american_scientific_journal-8025 110 36 ) ) PUNCT american_scientific_journal-8025 110 37 + + CCONJ american_scientific_journal-8025 110 38 𝐹(π‘š 𝐹(π‘š X american_scientific_journal-8025 110 39 + + NUM american_scientific_journal-8025 110 40 𝑒 𝑒 X american_scientific_journal-8025 110 41 ) ) PUNCT american_scientific_journal-8025 110 42 ≀ ≀ NUM american_scientific_journal-8025 110 43 1 1 NUM american_scientific_journal-8025 110 44 , , PUNCT american_scientific_journal-8025 110 45 but but CCONJ american_scientific_journal-8025 110 46 this this DET american_scientific_journal-8025 110 47 condition condition NOUN american_scientific_journal-8025 110 48 is be AUX american_scientific_journal-8025 110 49 nor nor CCONJ american_scientific_journal-8025 110 50 related relate VERB american_scientific_journal-8025 110 51 with with ADP american_scientific_journal-8025 110 52 skewness skewness NOUN american_scientific_journal-8025 110 53 but but CCONJ american_scientific_journal-8025 110 54 instead instead ADV american_scientific_journal-8025 110 55 a a DET american_scientific_journal-8025 110 56 relation relation NOUN american_scientific_journal-8025 110 57 in in ADP american_scientific_journal-8025 110 58 the the DET american_scientific_journal-8025 110 59 cumulative cumulative ADJ american_scientific_journal-8025 110 60 distribution distribution NOUN american_scientific_journal-8025 110 61 function function NOUN american_scientific_journal-8025 110 62 . . PUNCT american_scientific_journal-8025 111 1 2.2 2.2 NUM american_scientific_journal-8025 111 2 . . PUNCT american_scientific_journal-8025 112 1 variation variation NOUN american_scientific_journal-8025 112 2 coefficient coefficient NOUN american_scientific_journal-8025 112 3 . . PUNCT american_scientific_journal-8025 113 1 as as ADP american_scientific_journal-8025 113 2 in in ADP american_scientific_journal-8025 113 3 the the DET american_scientific_journal-8025 113 4 case case NOUN american_scientific_journal-8025 113 5 of of ADP american_scientific_journal-8025 113 6 skewness skewness NOUN american_scientific_journal-8025 113 7 , , PUNCT american_scientific_journal-8025 113 8 if if SCONJ american_scientific_journal-8025 113 9 the the DET american_scientific_journal-8025 113 10 agent agent NOUN american_scientific_journal-8025 113 11 is be AUX american_scientific_journal-8025 113 12 risk risk NOUN american_scientific_journal-8025 113 13 - - PUNCT american_scientific_journal-8025 113 14 averse averse ADJ american_scientific_journal-8025 113 15 , , PUNCT american_scientific_journal-8025 113 16 they they PRON american_scientific_journal-8025 113 17 perceive perceive VERB american_scientific_journal-8025 113 18 more more ADV american_scientific_journal-8025 113 19 the the DET american_scientific_journal-8025 113 20 risk risk NOUN american_scientific_journal-8025 113 21 for for ADP american_scientific_journal-8025 113 22 marginal marginal ADJ american_scientific_journal-8025 113 23 changes change NOUN american_scientific_journal-8025 113 24 close close ADJ american_scientific_journal-8025 113 25 to to ADP american_scientific_journal-8025 113 26 the the DET american_scientific_journal-8025 113 27 reference reference NOUN american_scientific_journal-8025 113 28 point point NOUN american_scientific_journal-8025 113 29 than than ADP american_scientific_journal-8025 113 30 those those PRON american_scientific_journal-8025 113 31 who who PRON american_scientific_journal-8025 113 32 are be AUX american_scientific_journal-8025 113 33 further far ADV american_scientific_journal-8025 113 34 away away ADV american_scientific_journal-8025 113 35 . . PUNCT american_scientific_journal-8025 114 1 in in ADP american_scientific_journal-8025 114 2 that that DET american_scientific_journal-8025 114 3 sense sense NOUN american_scientific_journal-8025 114 4 , , PUNCT american_scientific_journal-8025 114 5 there there PRON american_scientific_journal-8025 114 6 is be VERB american_scientific_journal-8025 114 7 a a DET american_scientific_journal-8025 114 8 difference difference NOUN american_scientific_journal-8025 114 9 with with ADP american_scientific_journal-8025 114 10 the the DET american_scientific_journal-8025 114 11 capm capm PROPN american_scientific_journal-8025 114 12 model model NOUN american_scientific_journal-8025 114 13 since since SCONJ american_scientific_journal-8025 114 14 the the DET american_scientific_journal-8025 114 15 variance variance NOUN american_scientific_journal-8025 114 16 gives give VERB american_scientific_journal-8025 114 17 greater great ADJ american_scientific_journal-8025 114 18 weight weight NOUN american_scientific_journal-8025 114 19 to to PART american_scientific_journal-8025 114 20 points point NOUN american_scientific_journal-8025 114 21 far far ADV american_scientific_journal-8025 114 22 from from ADP american_scientific_journal-8025 114 23 the the DET american_scientific_journal-8025 114 24 average average NOUN american_scientific_journal-8025 114 25 . . PUNCT american_scientific_journal-8025 115 1 this this PRON american_scientific_journal-8025 115 2 would would AUX american_scientific_journal-8025 115 3 correspond correspond VERB american_scientific_journal-8025 115 4 to to ADP american_scientific_journal-8025 115 5 agents agent NOUN american_scientific_journal-8025 115 6 whose whose DET american_scientific_journal-8025 115 7 distortion distortion NOUN american_scientific_journal-8025 115 8 function function NOUN american_scientific_journal-8025 115 9 is be AUX american_scientific_journal-8025 115 10 convex convex ADJ american_scientific_journal-8025 115 11 . . PUNCT american_scientific_journal-8025 116 1 some some DET american_scientific_journal-8025 116 2 properties property NOUN american_scientific_journal-8025 116 3 of of ADP american_scientific_journal-8025 116 4 the the DET american_scientific_journal-8025 116 5 variation variation NOUN american_scientific_journal-8025 116 6 coefficient coefficient NOUN american_scientific_journal-8025 116 7 are be AUX american_scientific_journal-8025 116 8 : : PUNCT american_scientific_journal-8025 116 9 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ NOUN american_scientific_journal-8025 116 10 ) ) PUNCT american_scientific_journal-8025 116 11 β‰₯ β‰₯ NOUN american_scientific_journal-8025 116 12 0 0 NUM american_scientific_journal-8025 116 13 , , PUNCT american_scientific_journal-8025 116 14 and and CCONJ american_scientific_journal-8025 116 15 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ NOUN american_scientific_journal-8025 116 16 ) ) PUNCT american_scientific_journal-8025 116 17 = = SYM american_scientific_journal-8025 116 18 0 0 PUNCT american_scientific_journal-8025 117 1 if if SCONJ american_scientific_journal-8025 117 2 and and CCONJ american_scientific_journal-8025 117 3 only only ADV american_scientific_journal-8025 117 4 if if SCONJ american_scientific_journal-8025 117 5 π‘₯ π‘₯ NOUN american_scientific_journal-8025 117 6 is be AUX american_scientific_journal-8025 117 7 constant constant ADJ american_scientific_journal-8025 117 8 . . PUNCT american_scientific_journal-8025 118 1 1 1 X american_scientific_journal-8025 118 2 . . X american_scientific_journal-8025 118 3 if if SCONJ american_scientific_journal-8025 118 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 118 5 ) ) PUNCT american_scientific_journal-8025 118 6 = = SYM american_scientific_journal-8025 118 7 π‘₯2 π‘₯2 NOUN american_scientific_journal-8025 118 8 then then ADV american_scientific_journal-8025 118 9 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ VERB american_scientific_journal-8025 118 10 ) ) PUNCT american_scientific_journal-8025 118 11 corresponds correspond VERB american_scientific_journal-8025 118 12 with with ADP american_scientific_journal-8025 118 13 the the DET american_scientific_journal-8025 118 14 variance variance NOUN american_scientific_journal-8025 118 15 . . PUNCT american_scientific_journal-8025 119 1 2 2 X american_scientific_journal-8025 119 2 . . X american_scientific_journal-8025 119 3 if if SCONJ american_scientific_journal-8025 119 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 119 5 ) ) PUNCT american_scientific_journal-8025 119 6 = = SYM american_scientific_journal-8025 120 1 π‘₯ π‘₯ PROPN american_scientific_journal-8025 120 2 then then ADV american_scientific_journal-8025 120 3 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ VERB american_scientific_journal-8025 120 4 ) ) PUNCT american_scientific_journal-8025 120 5 is be AUX american_scientific_journal-8025 120 6 corresponds correspond NOUN american_scientific_journal-8025 120 7 with with ADP american_scientific_journal-8025 120 8 the the DET american_scientific_journal-8025 120 9 mean mean ADJ american_scientific_journal-8025 120 10 absolute absolute ADJ american_scientific_journal-8025 120 11 deviation deviation NOUN american_scientific_journal-8025 120 12 𝑀𝐴𝐷. 𝑀𝐴𝐷. PUNCT american_scientific_journal-8025 120 13 3 3 X american_scientific_journal-8025 120 14 . . PUNCT american_scientific_journal-8025 120 15 by by ADP american_scientific_journal-8025 120 16 jensen jensen PROPN american_scientific_journal-8025 120 17 inequality inequality PROPN american_scientific_journal-8025 120 18 , , PUNCT american_scientific_journal-8025 120 19 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ PROPN american_scientific_journal-8025 120 20 ) ) PUNCT american_scientific_journal-8025 120 21 β‰₯ β‰₯ NOUN american_scientific_journal-8025 120 22 𝑓(𝑀𝐴𝐷(π‘₯ 𝑓(𝑀𝐴𝐷(π‘₯ NOUN american_scientific_journal-8025 120 23 ) ) PUNCT american_scientific_journal-8025 120 24 ) ) PUNCT american_scientific_journal-8025 121 1 if if SCONJ american_scientific_journal-8025 121 2 𝑓 𝑓 PRON american_scientific_journal-8025 121 3 is be AUX american_scientific_journal-8025 121 4 odd odd ADJ american_scientific_journal-8025 121 5 - - PUNCT american_scientific_journal-8025 121 6 convex convex NOUN american_scientific_journal-8025 121 7 , , PUNCT american_scientific_journal-8025 121 8 and and CCONJ american_scientific_journal-8025 121 9 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ NOUN american_scientific_journal-8025 121 10 ) ) PUNCT american_scientific_journal-8025 121 11 ≀ ≀ NUM american_scientific_journal-8025 121 12 𝑓(𝑀𝐴𝐷(π‘₯ 𝑓(𝑀𝐴𝐷(π‘₯ NOUN american_scientific_journal-8025 121 13 ) ) PUNCT american_scientific_journal-8025 121 14 ) ) PUNCT american_scientific_journal-8025 122 1 if if SCONJ american_scientific_journal-8025 122 2 𝑓 𝑓 PRON american_scientific_journal-8025 122 3 is be AUX american_scientific_journal-8025 122 4 odd odd ADV american_scientific_journal-8025 122 5 - - PUNCT american_scientific_journal-8025 122 6 concave concave NOUN american_scientific_journal-8025 122 7 . . PUNCT american_scientific_journal-8025 123 1 4 4 X american_scientific_journal-8025 123 2 . . X american_scientific_journal-8025 123 3 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ NOUN american_scientific_journal-8025 123 4 ) ) PUNCT american_scientific_journal-8025 123 5 β‰₯ β‰₯ NOUN american_scientific_journal-8025 123 6 |π‘†π‘˜π‘“(π‘₯)| |π‘†π‘˜π‘“(π‘₯)| PROPN american_scientific_journal-8025 123 7 5 5 NUM american_scientific_journal-8025 123 8 . . PUNCT american_scientific_journal-8025 124 1 if if SCONJ american_scientific_journal-8025 124 2 𝑓 𝑓 PRON american_scientific_journal-8025 124 3 is be AUX american_scientific_journal-8025 124 4 odd odd ADV american_scientific_journal-8025 124 5 - - PUNCT american_scientific_journal-8025 124 6 concave concave ADJ american_scientific_journal-8025 124 7 , , PUNCT american_scientific_journal-8025 124 8 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ NOUN american_scientific_journal-8025 124 9 ) ) PUNCT american_scientific_journal-8025 124 10 satisfies satisfie NOUN american_scientific_journal-8025 124 11 triangular triangular NOUN american_scientific_journal-8025 124 12 inequality inequality NOUN american_scientific_journal-8025 124 13 , , PUNCT american_scientific_journal-8025 124 14 i.e. i.e. X american_scientific_journal-8025 124 15 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ NOUN american_scientific_journal-8025 124 16 + + NOUN american_scientific_journal-8025 124 17 𝑦 𝑦 NOUN american_scientific_journal-8025 124 18 ) ) PUNCT american_scientific_journal-8025 124 19 ≀ ≀ NUM american_scientific_journal-8025 124 20 π‘‰π‘Žπ‘“(π‘₯ π‘‰π‘Žπ‘“(π‘₯ NOUN american_scientific_journal-8025 124 21 ) ) PUNCT american_scientific_journal-8025 124 22 + + CCONJ american_scientific_journal-8025 124 23 π‘‰π‘Žπ‘“(𝑦 π‘‰π‘Žπ‘“(𝑦 PROPN american_scientific_journal-8025 124 24 ) ) PUNCT american_scientific_journal-8025 124 25 . . PUNCT american_scientific_journal-8025 125 1 on on ADP american_scientific_journal-8025 125 2 other other ADJ american_scientific_journal-8025 125 3 side side NOUN american_scientific_journal-8025 125 4 π‘‰π‘Žπ‘“(Ξ±π‘₯ π‘‰π‘Žπ‘“(Ξ±π‘₯ NOUN american_scientific_journal-8025 125 5 ) ) PUNCT american_scientific_journal-8025 125 6 ≀ ≀ NUM american_scientific_journal-8025 125 7 |π‘Ž|π‘‰π‘Žπ‘“(π‘₯ |π‘Ž|π‘‰π‘Žπ‘“(π‘₯ PROPN american_scientific_journal-8025 125 8 ) ) PUNCT american_scientific_journal-8025 125 9 if if SCONJ american_scientific_journal-8025 125 10 Ξ± Ξ± PRON american_scientific_journal-8025 125 11 β‰₯ β‰₯ NOUN american_scientific_journal-8025 125 12 1 1 NUM american_scientific_journal-8025 125 13 , , PUNCT american_scientific_journal-8025 125 14 and and CCONJ american_scientific_journal-8025 125 15 π‘‰π‘Žπ‘“(Ξ±π‘₯ π‘‰π‘Žπ‘“(Ξ±π‘₯ NOUN american_scientific_journal-8025 125 16 ) ) PUNCT american_scientific_journal-8025 125 17 β‰₯ β‰₯ NOUN american_scientific_journal-8025 125 18 |π‘Ž|π‘‰π‘Žπ‘“(π‘₯ |π‘Ž|π‘‰π‘Žπ‘“(π‘₯ NOUN american_scientific_journal-8025 125 19 ) ) PUNCT american_scientific_journal-8025 125 20 if if SCONJ american_scientific_journal-8025 125 21 Ξ± Ξ± PRON american_scientific_journal-8025 125 22 < < X american_scientific_journal-8025 125 23 1 1 NUM american_scientific_journal-8025 125 24 . . PUNCT american_scientific_journal-8025 125 25 remark remark PROPN american_scientific_journal-8025 125 26 15 15 NUM american_scientific_journal-8025 125 27 : : PUNCT american_scientific_journal-8025 125 28 let let AUX american_scientific_journal-8025 125 29 be be AUX american_scientific_journal-8025 125 30 𝑋 𝑋 NOUN american_scientific_journal-8025 125 31 and and CCONJ american_scientific_journal-8025 125 32 π‘Œ π‘Œ PROPN american_scientific_journal-8025 125 33 two two NUM american_scientific_journal-8025 125 34 random random ADJ american_scientific_journal-8025 125 35 variables variable NOUN american_scientific_journal-8025 125 36 with with ADP american_scientific_journal-8025 125 37 distribution distribution NOUN american_scientific_journal-8025 125 38 functions function NOUN american_scientific_journal-8025 125 39 𝐹 𝐹 PROPN american_scientific_journal-8025 125 40 and and CCONJ american_scientific_journal-8025 125 41 𝐺 𝐺 PROPN american_scientific_journal-8025 125 42 respectively respectively ADV american_scientific_journal-8025 125 43 , , PUNCT american_scientific_journal-8025 125 44 then then ADV american_scientific_journal-8025 125 45 for for ADP american_scientific_journal-8025 125 46 all all DET american_scientific_journal-8025 125 47 𝑓 𝑓 DET american_scientific_journal-8025 125 48 odd odd ADJ american_scientific_journal-8025 125 49 and and CCONJ american_scientific_journal-8025 125 50 increasing increase VERB american_scientific_journal-8025 125 51 function function NOUN american_scientific_journal-8025 125 52 , , PUNCT american_scientific_journal-8025 125 53 π‘‰π‘Žπ‘“(𝑋 π‘‰π‘Žπ‘“(𝑋 NUM american_scientific_journal-8025 125 54 ) ) PUNCT american_scientific_journal-8025 125 55 β‰₯ β‰₯ NOUN american_scientific_journal-8025 125 56 π‘‰π‘Žπ‘“(π‘Œ π‘‰π‘Žπ‘“(π‘Œ NOUN american_scientific_journal-8025 125 57 ) ) PUNCT american_scientific_journal-8025 125 58 if if SCONJ american_scientific_journal-8025 125 59 and and CCONJ american_scientific_journal-8025 125 60 only only ADV american_scientific_journal-8025 125 61 if if SCONJ american_scientific_journal-8025 125 62 for for ADP american_scientific_journal-8025 125 63 all all DET american_scientific_journal-8025 125 64 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-8025 125 65 > > PUNCT american_scientific_journal-8025 125 66 0 0 PUNCT american_scientific_journal-8025 126 1 , , PUNCT american_scientific_journal-8025 126 2 𝑃(|𝑋 𝑃(|𝑋 PROPN american_scientific_journal-8025 126 3 βˆ’ βˆ’ PROPN american_scientific_journal-8025 126 4 𝐸𝑋| 𝐸𝑋| PROPN american_scientific_journal-8025 126 5 < < X american_scientific_journal-8025 126 6 π‘Ÿ π‘Ÿ X american_scientific_journal-8025 126 7 ) ) PUNCT american_scientific_journal-8025 126 8 ≀ ≀ NUM american_scientific_journal-8025 126 9 𝑃(|π‘Œ 𝑃(|π‘Œ PROPN american_scientific_journal-8025 126 10 βˆ’ βˆ’ PROPN american_scientific_journal-8025 127 1 πΈπ‘Œ| πΈπ‘Œ| PROPN american_scientific_journal-8025 127 2 < < X american_scientific_journal-8025 127 3 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-8025 127 4 ) ) PUNCT american_scientific_journal-8025 127 5 . . PUNCT american_scientific_journal-8025 128 1 that that PRON american_scientific_journal-8025 128 2 corresponds correspond VERB american_scientific_journal-8025 128 3 to to ADP american_scientific_journal-8025 128 4 the the DET american_scientific_journal-8025 128 5 " " PUNCT american_scientific_journal-8025 128 6 dispersion dispersion NOUN american_scientific_journal-8025 128 7 order order NOUN american_scientific_journal-8025 128 8 " " PUNCT american_scientific_journal-8025 128 9 notion notion NOUN american_scientific_journal-8025 128 10 introduced introduce VERB american_scientific_journal-8025 128 11 in in ADP american_scientific_journal-8025 128 12 [ [ X american_scientific_journal-8025 128 13 2 2 NUM american_scientific_journal-8025 128 14 ] ] PUNCT american_scientific_journal-8025 128 15 . . PUNCT american_scientific_journal-8025 129 1 2.3 2.3 NUM american_scientific_journal-8025 129 2 . . PUNCT american_scientific_journal-8025 129 3 relation relation NOUN american_scientific_journal-8025 129 4 with with ADP american_scientific_journal-8025 129 5 behavioral behavioral ADJ american_scientific_journal-8025 129 6 economics economic NOUN american_scientific_journal-8025 129 7 . . PUNCT american_scientific_journal-8025 130 1 consider consider VERB american_scientific_journal-8025 130 2 agents agent NOUN american_scientific_journal-8025 130 3 that that PRON american_scientific_journal-8025 130 4 behave behave VERB american_scientific_journal-8025 130 5 according accord VERB american_scientific_journal-8025 130 6 to to ADP american_scientific_journal-8025 130 7 prospect prospect NOUN american_scientific_journal-8025 130 8 theory theory NOUN american_scientific_journal-8025 130 9 and and CCONJ american_scientific_journal-8025 130 10 their their PRON american_scientific_journal-8025 130 11 reference reference NOUN american_scientific_journal-8025 130 12 point point NOUN american_scientific_journal-8025 130 13 is be AUX american_scientific_journal-8025 130 14 endogenous endogenous ADJ american_scientific_journal-8025 130 15 , , PUNCT american_scientific_journal-8025 130 16 similar similar ADJ american_scientific_journal-8025 130 17 to to ADP american_scientific_journal-8025 130 18 [ [ X american_scientific_journal-8025 130 19 3,8,5 3,8,5 NUM american_scientific_journal-8025 130 20 ] ] X american_scientific_journal-8025 130 21 . . PUNCT american_scientific_journal-8025 131 1 the the DET american_scientific_journal-8025 131 2 value value NOUN american_scientific_journal-8025 131 3 function function NOUN american_scientific_journal-8025 131 4 is be AUX american_scientific_journal-8025 131 5 generally generally ADV american_scientific_journal-8025 131 6 considered consider VERB american_scientific_journal-8025 131 7 concave concave VERB american_scientific_journal-8025 131 8 for for ADP american_scientific_journal-8025 131 9 gains gain NOUN american_scientific_journal-8025 131 10 and and CCONJ american_scientific_journal-8025 131 11 convex convex NOUN american_scientific_journal-8025 131 12 for for ADP american_scientific_journal-8025 131 13 losses loss NOUN american_scientific_journal-8025 131 14 , , PUNCT american_scientific_journal-8025 131 15 and and CCONJ american_scientific_journal-8025 131 16 even even ADV american_scientific_journal-8025 131 17 more more ADV american_scientific_journal-8025 131 18 [ [ X american_scientific_journal-8025 131 19 7 7 NUM american_scientific_journal-8025 131 20 ] ] PUNCT american_scientific_journal-8025 131 21 , , PUNCT american_scientific_journal-8025 131 22 american american ADJ american_scientific_journal-8025 131 23 academic academic ADJ american_scientific_journal-8025 131 24 scientific scientific ADJ american_scientific_journal-8025 131 25 research research NOUN american_scientific_journal-8025 131 26 journal journal NOUN american_scientific_journal-8025 131 27 for for ADP american_scientific_journal-8025 131 28 engineering engineering NOUN american_scientific_journal-8025 131 29 , , PUNCT american_scientific_journal-8025 131 30 technology technology NOUN american_scientific_journal-8025 131 31 , , PUNCT american_scientific_journal-8025 131 32 and and CCONJ american_scientific_journal-8025 131 33 sciences science NOUN american_scientific_journal-8025 131 34 ( ( PUNCT american_scientific_journal-8025 131 35 asrjets asrjet NOUN american_scientific_journal-8025 131 36 ) ) PUNCT american_scientific_journal-8025 131 37 ( ( PUNCT american_scientific_journal-8025 131 38 2022 2022 NUM american_scientific_journal-8025 131 39 ) ) PUNCT american_scientific_journal-8025 131 40 volume volume NOUN american_scientific_journal-8025 131 41 90 90 NUM american_scientific_journal-8025 131 42 , , PUNCT american_scientific_journal-8025 131 43 no1 no1 PROPN american_scientific_journal-8025 131 44 , , PUNCT american_scientific_journal-8025 131 45 pp pp ADV american_scientific_journal-8025 131 46 204 204 NUM american_scientific_journal-8025 131 47 - - SYM american_scientific_journal-8025 131 48 213 213 NUM american_scientific_journal-8025 131 49 212 212 NUM american_scientific_journal-8025 131 50 [ [ SYM american_scientific_journal-8025 131 51 6 6 NUM american_scientific_journal-8025 131 52 ] ] PUNCT american_scientific_journal-8025 131 53 described describe VERB american_scientific_journal-8025 131 54 how how SCONJ american_scientific_journal-8025 131 55 the the DET american_scientific_journal-8025 131 56 negative negative ADJ american_scientific_journal-8025 131 57 side side NOUN american_scientific_journal-8025 131 58 is be AUX american_scientific_journal-8025 131 59 in in ADP american_scientific_journal-8025 131 60 some some DET american_scientific_journal-8025 131 61 way way NOUN american_scientific_journal-8025 131 62 the the DET american_scientific_journal-8025 131 63 reflection reflection NOUN american_scientific_journal-8025 131 64 of of ADP american_scientific_journal-8025 131 65 the the DET american_scientific_journal-8025 131 66 positive positive ADJ american_scientific_journal-8025 131 67 one one NOUN american_scientific_journal-8025 131 68 . . PUNCT american_scientific_journal-8025 132 1 the the DET american_scientific_journal-8025 132 2 losses loss NOUN american_scientific_journal-8025 132 3 , , PUNCT american_scientific_journal-8025 132 4 in in ADP american_scientific_journal-8025 132 5 general general ADJ american_scientific_journal-8025 132 6 , , PUNCT american_scientific_journal-8025 132 7 correspond correspond VERB american_scientific_journal-8025 132 8 to to ADP american_scientific_journal-8025 132 9 the the DET american_scientific_journal-8025 132 10 reflection reflection NOUN american_scientific_journal-8025 132 11 of of ADP american_scientific_journal-8025 132 12 the the DET american_scientific_journal-8025 132 13 gains gain NOUN american_scientific_journal-8025 132 14 with with ADP american_scientific_journal-8025 132 15 respect respect NOUN american_scientific_journal-8025 132 16 to to ADP american_scientific_journal-8025 132 17 the the DET american_scientific_journal-8025 132 18 straight straight ADJ american_scientific_journal-8025 132 19 line line NOUN american_scientific_journal-8025 132 20 𝑦 𝑦 NOUN american_scientific_journal-8025 132 21 = = X american_scientific_journal-8025 132 22 βˆ’π‘₯ βˆ’π‘₯ NOUN american_scientific_journal-8025 132 23 multiplied multiply VERB american_scientific_journal-8025 132 24 by by ADP american_scientific_journal-8025 132 25 a a DET american_scientific_journal-8025 132 26 constant constant ADJ american_scientific_journal-8025 132 27 Ξ» Ξ» NOUN american_scientific_journal-8025 132 28 denominated denominate VERB american_scientific_journal-8025 132 29 loss loss NOUN american_scientific_journal-8025 132 30 aversion aversion NOUN american_scientific_journal-8025 132 31 coefficient coefficient NOUN american_scientific_journal-8025 132 32 , , PUNCT american_scientific_journal-8025 132 33 whose whose DET american_scientific_journal-8025 132 34 value value NOUN american_scientific_journal-8025 132 35 has have AUX american_scientific_journal-8025 132 36 been be AUX american_scientific_journal-8025 132 37 determined determine VERB american_scientific_journal-8025 132 38 to to PART american_scientific_journal-8025 132 39 be be AUX american_scientific_journal-8025 132 40 between between ADP american_scientific_journal-8025 132 41 1.5 1.5 NUM american_scientific_journal-8025 132 42 and and CCONJ american_scientific_journal-8025 132 43 5 5 NUM american_scientific_journal-8025 132 44 . . PUNCT american_scientific_journal-8025 133 1 if if SCONJ american_scientific_journal-8025 133 2 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 133 3 ) ) PUNCT american_scientific_journal-8025 133 4 represent represent VERB american_scientific_journal-8025 133 5 the the DET american_scientific_journal-8025 133 6 concave concave NOUN american_scientific_journal-8025 133 7 value value NOUN american_scientific_journal-8025 133 8 function function NOUN american_scientific_journal-8025 133 9 for for ADP american_scientific_journal-8025 133 10 gains gain NOUN american_scientific_journal-8025 133 11 then then ADV american_scientific_journal-8025 133 12 the the DET american_scientific_journal-8025 133 13 value value NOUN american_scientific_journal-8025 133 14 function function NOUN american_scientific_journal-8025 133 15 ΞΌ ΞΌ PROPN american_scientific_journal-8025 133 16 is be AUX american_scientific_journal-8025 133 17 equivalent equivalent ADJ american_scientific_journal-8025 133 18 to to ADP american_scientific_journal-8025 133 19 ΞΌ(π‘₯ ΞΌ(π‘₯ PROPN american_scientific_journal-8025 133 20 ) ) PUNCT american_scientific_journal-8025 134 1 = = SYM american_scientific_journal-8025 134 2 min(𝑓(π‘₯ min(𝑓(π‘₯ PROPN american_scientific_journal-8025 134 3 ) ) PUNCT american_scientific_journal-8025 134 4 , , PUNCT american_scientific_journal-8025 134 5 λ𝑓(π‘₯ λ𝑓(π‘₯ NOUN american_scientific_journal-8025 134 6 ) ) PUNCT american_scientific_journal-8025 134 7 ) ) PUNCT american_scientific_journal-8025 135 1 = = SYM american_scientific_journal-8025 135 2 1 1 NUM american_scientific_journal-8025 135 3 2 2 NUM american_scientific_journal-8025 135 4 ( ( PUNCT american_scientific_journal-8025 135 5 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-8025 135 6 ) ) PUNCT american_scientific_journal-8025 135 7 + + NUM american_scientific_journal-8025 135 8 λ𝑓(π‘₯ λ𝑓(π‘₯ NOUN american_scientific_journal-8025 135 9 ) ) PUNCT american_scientific_journal-8025 135 10 βˆ’ βˆ’ PROPN american_scientific_journal-8025 135 11 |𝑓(π‘₯ |𝑓(π‘₯ PROPN american_scientific_journal-8025 135 12 ) ) PUNCT american_scientific_journal-8025 135 13 βˆ’ βˆ’ NOUN american_scientific_journal-8025 135 14 λ𝑓(π‘₯)| λ𝑓(π‘₯)| NOUN american_scientific_journal-8025 135 15 ) ) PUNCT american_scientific_journal-8025 135 16 ( ( PUNCT american_scientific_journal-8025 135 17 20 20 NUM american_scientific_journal-8025 135 18 ) ) PUNCT american_scientific_journal-8025 135 19 if if SCONJ american_scientific_journal-8025 135 20 we we PRON american_scientific_journal-8025 135 21 consider consider VERB american_scientific_journal-8025 135 22 that that SCONJ american_scientific_journal-8025 135 23 outcomes outcome NOUN american_scientific_journal-8025 135 24 have have VERB american_scientific_journal-8025 135 25 a a DET american_scientific_journal-8025 135 26 distribution distribution NOUN american_scientific_journal-8025 135 27 𝐹 𝐹 PROPN american_scientific_journal-8025 135 28 and and CCONJ american_scientific_journal-8025 135 29 the the DET american_scientific_journal-8025 135 30 endogenous endogenous ADJ american_scientific_journal-8025 135 31 reference reference NOUN american_scientific_journal-8025 135 32 point point NOUN american_scientific_journal-8025 135 33 is be AUX american_scientific_journal-8025 135 34 given give VERB american_scientific_journal-8025 135 35 by by ADP american_scientific_journal-8025 135 36 𝐸(𝑋 𝐸(𝑋 NOUN american_scientific_journal-8025 135 37 ) ) PUNCT american_scientific_journal-8025 135 38 , , PUNCT american_scientific_journal-8025 135 39 ( ( PUNCT american_scientific_journal-8025 135 40 21 21 NUM american_scientific_journal-8025 135 41 ) ) PUNCT american_scientific_journal-8025 135 42 from from ADP american_scientific_journal-8025 135 43 this this DET american_scientific_journal-8025 135 44 representation representation NOUN american_scientific_journal-8025 135 45 and and CCONJ american_scientific_journal-8025 135 46 using use VERB american_scientific_journal-8025 135 47 the the DET american_scientific_journal-8025 135 48 properties property NOUN american_scientific_journal-8025 135 49 listed list VERB american_scientific_journal-8025 135 50 previously previously ADV american_scientific_journal-8025 135 51 in in ADP american_scientific_journal-8025 135 52 this this DET american_scientific_journal-8025 135 53 paper paper NOUN american_scientific_journal-8025 135 54 , , PUNCT american_scientific_journal-8025 135 55 it it PRON american_scientific_journal-8025 135 56 is be AUX american_scientific_journal-8025 135 57 possible possible ADJ american_scientific_journal-8025 135 58 to to PART american_scientific_journal-8025 135 59 measure measure VERB american_scientific_journal-8025 135 60 the the DET american_scientific_journal-8025 135 61 impact impact NOUN american_scientific_journal-8025 135 62 of of ADP american_scientific_journal-8025 135 63 agents agent NOUN american_scientific_journal-8025 135 64 as as SCONJ american_scientific_journal-8025 135 65 described describe VERB american_scientific_journal-8025 135 66 in in ADP american_scientific_journal-8025 135 67 real real ADJ american_scientific_journal-8025 135 68 economies economy NOUN american_scientific_journal-8025 135 69 . . PUNCT american_scientific_journal-8025 136 1 3 3 X american_scientific_journal-8025 136 2 . . X american_scientific_journal-8025 136 3 conclusions conclusion NOUN american_scientific_journal-8025 136 4 in in ADP american_scientific_journal-8025 136 5 this this DET american_scientific_journal-8025 136 6 paper paper NOUN american_scientific_journal-8025 136 7 , , PUNCT american_scientific_journal-8025 136 8 we we PRON american_scientific_journal-8025 136 9 showed show VERB american_scientific_journal-8025 136 10 that that SCONJ american_scientific_journal-8025 136 11 the the DET american_scientific_journal-8025 136 12 standard standard ADJ american_scientific_journal-8025 136 13 measure measure NOUN american_scientific_journal-8025 136 14 of of ADP american_scientific_journal-8025 136 15 skewness skewness NOUN american_scientific_journal-8025 136 16 could could AUX american_scientific_journal-8025 136 17 be be AUX american_scientific_journal-8025 136 18 extended extend VERB american_scientific_journal-8025 136 19 for for ADP american_scientific_journal-8025 136 20 the the DET american_scientific_journal-8025 136 21 class class NOUN american_scientific_journal-8025 136 22 of of ADP american_scientific_journal-8025 136 23 odd odd ADJ american_scientific_journal-8025 136 24 and and CCONJ american_scientific_journal-8025 136 25 increasing increase VERB american_scientific_journal-8025 136 26 functions function NOUN american_scientific_journal-8025 136 27 . . PUNCT american_scientific_journal-8025 137 1 a a DET american_scientific_journal-8025 137 2 more more ADV american_scientific_journal-8025 137 3 robust robust ADJ american_scientific_journal-8025 137 4 notion notion NOUN american_scientific_journal-8025 137 5 of of ADP american_scientific_journal-8025 137 6 asymmetry asymmetry NOUN american_scientific_journal-8025 137 7 can can AUX american_scientific_journal-8025 137 8 be be AUX american_scientific_journal-8025 137 9 defined define VERB american_scientific_journal-8025 137 10 and and CCONJ american_scientific_journal-8025 137 11 is be AUX american_scientific_journal-8025 137 12 consistent consistent ADJ american_scientific_journal-8025 137 13 with with ADP american_scientific_journal-8025 137 14 all all DET american_scientific_journal-8025 137 15 the the DET american_scientific_journal-8025 137 16 expected expect VERB american_scientific_journal-8025 137 17 skewness skewness NOUN american_scientific_journal-8025 137 18 properties property NOUN american_scientific_journal-8025 137 19 . . PUNCT american_scientific_journal-8025 138 1 we we PRON american_scientific_journal-8025 138 2 prove prove VERB american_scientific_journal-8025 138 3 that that SCONJ american_scientific_journal-8025 138 4 for for ADP american_scientific_journal-8025 138 5 strong strong ADJ american_scientific_journal-8025 138 6 asymmetric asymmetric ADJ american_scientific_journal-8025 138 7 distributions distribution NOUN american_scientific_journal-8025 138 8 , , PUNCT american_scientific_journal-8025 138 9 the the DET american_scientific_journal-8025 138 10 inequalities inequality NOUN american_scientific_journal-8025 138 11 π‘šπ‘’π‘‘π‘–π‘Žπ‘› π‘šπ‘’π‘‘π‘–π‘Žπ‘› VERB american_scientific_journal-8025 138 12 ≀ ≀ X american_scientific_journal-8025 138 13 π‘šπ‘’π‘Žπ‘›$ π‘šπ‘’π‘Žπ‘›$ NOUN american_scientific_journal-8025 138 14 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ american_scientific_journal-8025 138 15 $ $ SYM american_scientific_journal-8025 138 16 π‘šπ‘œπ‘‘π‘’ π‘šπ‘œπ‘‘π‘’ NOUN american_scientific_journal-8025 138 17 ≀ ≀ NUM american_scientific_journal-8025 138 18 π‘šπ‘’π‘Žπ‘› π‘šπ‘’π‘Žπ‘› NOUN american_scientific_journal-8025 138 19 are be AUX american_scientific_journal-8025 138 20 valid valid ADJ american_scientific_journal-8025 138 21 , , PUNCT american_scientific_journal-8025 138 22 but but CCONJ american_scientific_journal-8025 138 23 even even ADV american_scientific_journal-8025 138 24 in in ADP american_scientific_journal-8025 138 25 that that DET american_scientific_journal-8025 138 26 case case NOUN american_scientific_journal-8025 138 27 , , PUNCT american_scientific_journal-8025 138 28 there there PRON american_scientific_journal-8025 138 29 is be VERB american_scientific_journal-8025 138 30 no no DET american_scientific_journal-8025 138 31 relation relation NOUN american_scientific_journal-8025 138 32 between between ADP american_scientific_journal-8025 138 33 the the DET american_scientific_journal-8025 138 34 mode mode NOUN american_scientific_journal-8025 138 35 and and CCONJ american_scientific_journal-8025 138 36 the the DET american_scientific_journal-8025 138 37 median median NOUN american_scientific_journal-8025 138 38 . . PUNCT american_scientific_journal-8025 139 1 finally finally ADV american_scientific_journal-8025 139 2 , , PUNCT american_scientific_journal-8025 139 3 there there PRON american_scientific_journal-8025 139 4 is be VERB american_scientific_journal-8025 139 5 a a DET american_scientific_journal-8025 139 6 strong strong ADJ american_scientific_journal-8025 139 7 relationship relationship NOUN american_scientific_journal-8025 139 8 between between ADP american_scientific_journal-8025 139 9 the the DET american_scientific_journal-8025 139 10 skewness skewness NOUN american_scientific_journal-8025 139 11 and and CCONJ american_scientific_journal-8025 139 12 variation variation NOUN american_scientific_journal-8025 139 13 coefficients coefficient NOUN american_scientific_journal-8025 139 14 defined define VERB american_scientific_journal-8025 139 15 in in ADP american_scientific_journal-8025 139 16 this this DET american_scientific_journal-8025 139 17 paper paper NOUN american_scientific_journal-8025 139 18 with with ADP american_scientific_journal-8025 139 19 works work NOUN american_scientific_journal-8025 139 20 in in ADP american_scientific_journal-8025 139 21 regret regret NOUN american_scientific_journal-8025 139 22 , , PUNCT american_scientific_journal-8025 139 23 disappointment disappointment NOUN american_scientific_journal-8025 139 24 , , PUNCT american_scientific_journal-8025 139 25 and and CCONJ american_scientific_journal-8025 139 26 prospect prospect NOUN american_scientific_journal-8025 139 27 theory theory NOUN american_scientific_journal-8025 139 28 such such ADJ american_scientific_journal-8025 139 29 as as ADP american_scientific_journal-8025 139 30 [ [ X american_scientific_journal-8025 139 31 3,8,5 3,8,5 NUM american_scientific_journal-8025 139 32 ] ] PUNCT american_scientific_journal-8025 139 33 . . PUNCT american_scientific_journal-8025 140 1 references reference NOUN american_scientific_journal-8025 140 2 [ [ X american_scientific_journal-8025 140 3 1 1 NUM american_scientific_journal-8025 140 4 ] ] PUNCT american_scientific_journal-8025 140 5 abadir abadir NOUN american_scientific_journal-8025 140 6 , , PUNCT american_scientific_journal-8025 140 7 k. k. PROPN american_scientific_journal-8025 140 8 m. m. PROPN american_scientific_journal-8025 140 9 β€œ " PUNCT american_scientific_journal-8025 140 10 the the DET american_scientific_journal-8025 140 11 mean mean ADJ american_scientific_journal-8025 140 12 - - PUNCT american_scientific_journal-8025 140 13 median median ADJ american_scientific_journal-8025 140 14 - - PUNCT american_scientific_journal-8025 140 15 mode mode NOUN american_scientific_journal-8025 140 16 inequality inequality NOUN american_scientific_journal-8025 140 17 : : PUNCT american_scientific_journal-8025 140 18 counterexamples counterexample NOUN american_scientific_journal-8025 140 19 ” " PUNCT american_scientific_journal-8025 140 20 . . PUNCT american_scientific_journal-8025 141 1 econometric econometric PROPN american_scientific_journal-8025 141 2 theory theory NOUN american_scientific_journal-8025 141 3 21 21 NUM american_scientific_journal-8025 141 4 ( ( PUNCT american_scientific_journal-8025 141 5 2 2 NUM american_scientific_journal-8025 141 6 ) ) PUNCT american_scientific_journal-8025 141 7 , , PUNCT american_scientific_journal-8025 141 8 477–482 477–482 NUM american_scientific_journal-8025 141 9 . . NOUN american_scientific_journal-8025 141 10 2005 2005 NUM american_scientific_journal-8025 141 11 . . PUNCT american_scientific_journal-8025 142 1 [ [ X american_scientific_journal-8025 142 2 2 2 NUM american_scientific_journal-8025 142 3 ] ] X american_scientific_journal-8025 142 4 basu basu PROPN american_scientific_journal-8025 142 5 , , PUNCT american_scientific_journal-8025 142 6 s. s. PROPN american_scientific_journal-8025 142 7 and and CCONJ american_scientific_journal-8025 142 8 a. a. PROPN american_scientific_journal-8025 142 9 dasgupt dasgupt PROPN american_scientific_journal-8025 142 10 . . PUNCT american_scientific_journal-8025 143 1 β€œ " PUNCT american_scientific_journal-8025 143 2 the the DET american_scientific_journal-8025 143 3 mean mean ADJ american_scientific_journal-8025 143 4 , , PUNCT american_scientific_journal-8025 143 5 median median ADJ american_scientific_journal-8025 143 6 , , PUNCT american_scientific_journal-8025 143 7 and and CCONJ american_scientific_journal-8025 143 8 mode mode NOUN american_scientific_journal-8025 143 9 of of ADP american_scientific_journal-8025 143 10 unimodal unimodal ADJ american_scientific_journal-8025 143 11 distributions distribution NOUN american_scientific_journal-8025 143 12 : : PUNCT american_scientific_journal-8025 143 13 a a DET american_scientific_journal-8025 143 14 characterization characterization NOUN american_scientific_journal-8025 143 15 ” " PUNCT american_scientific_journal-8025 143 16 . . PUNCT american_scientific_journal-8025 144 1 theory theory NOUN american_scientific_journal-8025 144 2 of of ADP american_scientific_journal-8025 144 3 probability probability NOUN american_scientific_journal-8025 144 4 & & CCONJ american_scientific_journal-8025 144 5 its its PRON american_scientific_journal-8025 144 6 applications application NOUN american_scientific_journal-8025 144 7 41 41 NUM american_scientific_journal-8025 144 8 ( ( PUNCT american_scientific_journal-8025 144 9 2 2 NUM american_scientific_journal-8025 144 10 ) ) PUNCT american_scientific_journal-8025 144 11 , , PUNCT american_scientific_journal-8025 144 12 210–223 210–223 NUM american_scientific_journal-8025 144 13 . . PUNCT american_scientific_journal-8025 145 1 1995 1995 NUM american_scientific_journal-8025 146 1 [ [ X american_scientific_journal-8025 146 2 3 3 NUM american_scientific_journal-8025 146 3 ] ] X american_scientific_journal-8025 146 4 bell bell NOUN american_scientific_journal-8025 146 5 , , PUNCT american_scientific_journal-8025 146 6 d. d. PROPN american_scientific_journal-8025 146 7 e. e. PROPN american_scientific_journal-8025 147 1 β€œ " PUNCT american_scientific_journal-8025 147 2 disappointment disappointment NOUN american_scientific_journal-8025 147 3 in in ADP american_scientific_journal-8025 147 4 decision decision NOUN american_scientific_journal-8025 147 5 making make VERB american_scientific_journal-8025 147 6 under under ADP american_scientific_journal-8025 147 7 uncertainty uncertainty NOUN american_scientific_journal-8025 147 8 ” " PUNCT american_scientific_journal-8025 147 9 . . PUNCT american_scientific_journal-8025 148 1 operations operation NOUN american_scientific_journal-8025 148 2 research research VERB american_scientific_journal-8025 148 3 33 33 NUM american_scientific_journal-8025 148 4 ( ( PUNCT american_scientific_journal-8025 148 5 1 1 NUM american_scientific_journal-8025 148 6 ) ) PUNCT american_scientific_journal-8025 148 7 , , PUNCT american_scientific_journal-8025 148 8 1–27 1–27 PROPN american_scientific_journal-8025 148 9 . . NOUN american_scientific_journal-8025 148 10 1995 1995 NUM american_scientific_journal-8025 149 1 [ [ X american_scientific_journal-8025 149 2 4 4 NUM american_scientific_journal-8025 149 3 ] ] SYM american_scientific_journal-8025 149 4 bickel bickel PROPN american_scientific_journal-8025 149 5 , , PUNCT american_scientific_journal-8025 149 6 p. p. PROPN american_scientific_journal-8025 149 7 j. j. PROPN american_scientific_journal-8025 149 8 and and CCONJ american_scientific_journal-8025 149 9 e. e. PROPN american_scientific_journal-8025 149 10 l. l. PROPN american_scientific_journal-8025 149 11 lehmann lehmann PROPN american_scientific_journal-8025 149 12 . . PUNCT american_scientific_journal-8025 150 1 β€œ " PUNCT american_scientific_journal-8025 150 2 descriptive descriptive ADJ american_scientific_journal-8025 150 3 statistics statistic NOUN american_scientific_journal-8025 150 4 for for ADP american_scientific_journal-8025 150 5 nonparametric nonparametric NOUN american_scientific_journal-8025 150 6 models model NOUN american_scientific_journal-8025 150 7 . . PUNCT american_scientific_journal-8025 151 1 iii iii X american_scientific_journal-8025 151 2 . . PUNCT american_scientific_journal-8025 151 3 dispersion dispersion NOUN american_scientific_journal-8025 151 4 ” " PUNCT american_scientific_journal-8025 151 5 . . PUNCT american_scientific_journal-8025 152 1 the the DET american_scientific_journal-8025 152 2 american american PROPN american_scientific_journal-8025 152 3 academic academic PROPN american_scientific_journal-8025 152 4 scientific scientific ADJ american_scientific_journal-8025 152 5 research research NOUN american_scientific_journal-8025 152 6 journal journal NOUN american_scientific_journal-8025 152 7 for for ADP american_scientific_journal-8025 152 8 engineering engineering NOUN american_scientific_journal-8025 152 9 , , PUNCT american_scientific_journal-8025 152 10 technology technology NOUN american_scientific_journal-8025 152 11 , , PUNCT american_scientific_journal-8025 152 12 and and CCONJ american_scientific_journal-8025 152 13 sciences science NOUN american_scientific_journal-8025 152 14 ( ( PUNCT american_scientific_journal-8025 152 15 asrjets asrjet NOUN american_scientific_journal-8025 152 16 ) ) PUNCT american_scientific_journal-8025 152 17 ( ( PUNCT american_scientific_journal-8025 152 18 2022 2022 NUM american_scientific_journal-8025 152 19 ) ) PUNCT american_scientific_journal-8025 152 20 volume volume NOUN american_scientific_journal-8025 152 21 90 90 NUM american_scientific_journal-8025 152 22 , , PUNCT american_scientific_journal-8025 152 23 no1 no1 PROPN american_scientific_journal-8025 152 24 , , PUNCT american_scientific_journal-8025 152 25 pp pp ADV american_scientific_journal-8025 152 26 204 204 NUM american_scientific_journal-8025 152 27 - - SYM american_scientific_journal-8025 152 28 213 213 NUM american_scientific_journal-8025 152 29 213 213 NUM american_scientific_journal-8025 152 30 annals annal NOUN american_scientific_journal-8025 152 31 of of ADP american_scientific_journal-8025 152 32 statistics statistic NOUN american_scientific_journal-8025 152 33 4 4 NUM american_scientific_journal-8025 152 34 ( ( PUNCT american_scientific_journal-8025 152 35 6 6 NUM american_scientific_journal-8025 152 36 ) ) PUNCT american_scientific_journal-8025 152 37 , , PUNCT american_scientific_journal-8025 152 38 1139–1158 1139–1158 NUM american_scientific_journal-8025 152 39 . . PUNCT american_scientific_journal-8025 153 1 1976 1976 NUM american_scientific_journal-8025 153 2 [ [ X american_scientific_journal-8025 153 3 5 5 NUM american_scientific_journal-8025 153 4 ] ] X american_scientific_journal-8025 153 5 gul gul PROPN american_scientific_journal-8025 153 6 , , PUNCT american_scientific_journal-8025 153 7 f. f. PROPN american_scientific_journal-8025 153 8 β€œ " PUNCT american_scientific_journal-8025 153 9 a a DET american_scientific_journal-8025 153 10 theory theory NOUN american_scientific_journal-8025 153 11 of of ADP american_scientific_journal-8025 153 12 disappointment disappointment ADJ american_scientific_journal-8025 153 13 aversion aversion NOUN american_scientific_journal-8025 153 14 ” " PUNCT american_scientific_journal-8025 153 15 . . PUNCT american_scientific_journal-8025 154 1 econometrica econometrica PROPN american_scientific_journal-8025 154 2 : : PUNCT american_scientific_journal-8025 154 3 journal journal NOUN american_scientific_journal-8025 154 4 of of ADP american_scientific_journal-8025 154 5 the the DET american_scientific_journal-8025 154 6 conometric conometric ADJ american_scientific_journal-8025 154 7 society society NOUN american_scientific_journal-8025 154 8 , , PUNCT american_scientific_journal-8025 154 9 667 667 NUM american_scientific_journal-8025 154 10 – – PUNCT american_scientific_journal-8025 154 11 686 686 NUM american_scientific_journal-8025 154 12 . . X american_scientific_journal-8025 154 13 1991 1991 NUM american_scientific_journal-8025 155 1 [ [ X american_scientific_journal-8025 155 2 6 6 NUM american_scientific_journal-8025 155 3 ] ] PUNCT american_scientific_journal-8025 155 4 kahneman kahneman NOUN american_scientific_journal-8025 155 5 , , PUNCT american_scientific_journal-8025 155 6 d. d. PROPN american_scientific_journal-8025 155 7 , , PUNCT american_scientific_journal-8025 155 8 j. j. PROPN american_scientific_journal-8025 155 9 l. l. PROPN american_scientific_journal-8025 155 10 knetsch knetsch PROPN american_scientific_journal-8025 155 11 , , PUNCT american_scientific_journal-8025 155 12 and and CCONJ american_scientific_journal-8025 155 13 r. r. PROPN american_scientific_journal-8025 155 14 h. h. PROPN american_scientific_journal-8025 155 15 thaler thaler PROPN american_scientific_journal-8025 155 16 . . PUNCT american_scientific_journal-8025 156 1 β€œ " PUNCT american_scientific_journal-8025 156 2 anomalies anomaly NOUN american_scientific_journal-8025 156 3 : : PUNCT american_scientific_journal-8025 156 4 the the DET american_scientific_journal-8025 156 5 endowment endowment ADJ american_scientific_journal-8025 156 6 effect effect NOUN american_scientific_journal-8025 156 7 , , PUNCT american_scientific_journal-8025 156 8 loss loss NOUN american_scientific_journal-8025 156 9 aversion aversion NOUN american_scientific_journal-8025 156 10 , , PUNCT american_scientific_journal-8025 156 11 and and CCONJ american_scientific_journal-8025 156 12 status status NOUN american_scientific_journal-8025 156 13 quo quo NOUN american_scientific_journal-8025 156 14 bias bias NOUN american_scientific_journal-8025 156 15 ” " PUNCT american_scientific_journal-8025 156 16 . . PUNCT american_scientific_journal-8025 157 1 journal journal PROPN american_scientific_journal-8025 157 2 of of ADP american_scientific_journal-8025 157 3 economic economic ADJ american_scientific_journal-8025 157 4 perspectives perspective NOUN american_scientific_journal-8025 157 5 5 5 NUM american_scientific_journal-8025 157 6 ( ( PUNCT american_scientific_journal-8025 157 7 1 1 NUM american_scientific_journal-8025 157 8 ) ) PUNCT american_scientific_journal-8025 157 9 , , PUNCT american_scientific_journal-8025 157 10 193–206 193–206 NUM american_scientific_journal-8025 157 11 . . PUNCT american_scientific_journal-8025 157 12 1991 1991 NUM american_scientific_journal-8025 158 1 [ [ X american_scientific_journal-8025 158 2 7 7 X american_scientific_journal-8025 158 3 ] ] X american_scientific_journal-8025 158 4 kahneman kahneman NOUN american_scientific_journal-8025 158 5 , , PUNCT american_scientific_journal-8025 158 6 d. d. PROPN american_scientific_journal-8025 158 7 and and CCONJ american_scientific_journal-8025 158 8 a. a. NOUN american_scientific_journal-8025 158 9 tversky tversky PROPN american_scientific_journal-8025 158 10 . . PUNCT american_scientific_journal-8025 159 1 β€œ " PUNCT american_scientific_journal-8025 159 2 prospect prospect NOUN american_scientific_journal-8025 159 3 theory theory NOUN american_scientific_journal-8025 159 4 : : PUNCT american_scientific_journal-8025 159 5 an an DET american_scientific_journal-8025 159 6 analysis analysis NOUN american_scientific_journal-8025 159 7 of of ADP american_scientific_journal-8025 159 8 decision decision NOUN american_scientific_journal-8025 159 9 under under ADP american_scientific_journal-8025 159 10 risk risk NOUN american_scientific_journal-8025 159 11 ” " PUNCT american_scientific_journal-8025 159 12 . . PUNCT american_scientific_journal-8025 160 1 econometrica econometrica PROPN american_scientific_journal-8025 160 2 47 47 NUM american_scientific_journal-8025 160 3 ( ( PUNCT american_scientific_journal-8025 160 4 2 2 NUM american_scientific_journal-8025 160 5 ) ) PUNCT american_scientific_journal-8025 160 6 , , PUNCT american_scientific_journal-8025 160 7 263–292 263–292 NUM american_scientific_journal-8025 160 8 . . PUNCT american_scientific_journal-8025 160 9 1979 1979 NUM american_scientific_journal-8025 160 10 . . PUNCT american_scientific_journal-8025 161 1 [ [ X american_scientific_journal-8025 161 2 8 8 NUM american_scientific_journal-8025 161 3 ] ] PUNCT american_scientific_journal-8025 161 4 loomes loome NOUN american_scientific_journal-8025 161 5 , , PUNCT american_scientific_journal-8025 161 6 g. g. PROPN american_scientific_journal-8025 161 7 and and CCONJ american_scientific_journal-8025 161 8 r. r. PROPN american_scientific_journal-8025 161 9 sugden sugden PROPN american_scientific_journal-8025 161 10 . . PUNCT american_scientific_journal-8025 162 1 β€œ " PUNCT american_scientific_journal-8025 162 2 disappointment disappointment NOUN american_scientific_journal-8025 162 3 and and CCONJ american_scientific_journal-8025 162 4 dynamic dynamic ADJ american_scientific_journal-8025 162 5 consistency consistency NOUN american_scientific_journal-8025 162 6 in in ADP american_scientific_journal-8025 162 7 choice choice NOUN american_scientific_journal-8025 162 8 under under ADP american_scientific_journal-8025 162 9 uncertainty uncertainty NOUN american_scientific_journal-8025 162 10 ” " PUNCT american_scientific_journal-8025 162 11 . . PUNCT american_scientific_journal-8025 163 1 the the DET american_scientific_journal-8025 163 2 review review NOUN american_scientific_journal-8025 163 3 of of ADP american_scientific_journal-8025 163 4 economic economic ADJ american_scientific_journal-8025 163 5 studies study NOUN american_scientific_journal-8025 163 6 53 53 NUM american_scientific_journal-8025 163 7 ( ( PUNCT american_scientific_journal-8025 163 8 2 2 NUM american_scientific_journal-8025 163 9 ) ) PUNCT american_scientific_journal-8025 163 10 , , PUNCT american_scientific_journal-8025 163 11 271–282 271–282 NUM american_scientific_journal-8025 163 12 . . PUNCT american_scientific_journal-8025 163 13 1986 1986 NUM american_scientific_journal-8025 164 1 [ [ X american_scientific_journal-8025 164 2 9 9 NUM american_scientific_journal-8025 164 3 ] ] X american_scientific_journal-8025 164 4 oja oja NOUN american_scientific_journal-8025 164 5 , , PUNCT american_scientific_journal-8025 164 6 h. h. PROPN american_scientific_journal-8025 164 7 β€œ " PUNCT american_scientific_journal-8025 164 8 on on ADP american_scientific_journal-8025 164 9 location location NOUN american_scientific_journal-8025 164 10 , , PUNCT american_scientific_journal-8025 164 11 scale scale NOUN american_scientific_journal-8025 164 12 , , PUNCT american_scientific_journal-8025 164 13 skewness skewness NOUN american_scientific_journal-8025 164 14 and and CCONJ american_scientific_journal-8025 164 15 kurtosis kurtosis NOUN american_scientific_journal-8025 164 16 of of ADP american_scientific_journal-8025 164 17 univariate univariate ADJ american_scientific_journal-8025 164 18 distributions distribution NOUN american_scientific_journal-8025 164 19 ” " PUNCT american_scientific_journal-8025 164 20 . . PUNCT american_scientific_journal-8025 165 1 scandinavian scandinavian ADJ american_scientific_journal-8025 165 2 journal journal PROPN american_scientific_journal-8025 165 3 of of ADP american_scientific_journal-8025 165 4 statistics statistic NOUN american_scientific_journal-8025 165 5 , , PUNCT american_scientific_journal-8025 165 6 154–168 154–168 NUM american_scientific_journal-8025 165 7 . . PUNCT american_scientific_journal-8025 165 8 1981 1981 NUM american_scientific_journal-8025 166 1 [ [ X american_scientific_journal-8025 166 2 10 10 NUM american_scientific_journal-8025 166 3 ] ] SYM american_scientific_journal-8025 166 4 vanzwet vanzwet PROPN american_scientific_journal-8025 166 5 , , PUNCT american_scientific_journal-8025 166 6 w. w. PROPN american_scientific_journal-8025 166 7 r. r. PROPN american_scientific_journal-8025 166 8 β€œ " PUNCT american_scientific_journal-8025 166 9 convex convex NOUN american_scientific_journal-8025 166 10 transformations transformation NOUN american_scientific_journal-8025 166 11 of of ADP american_scientific_journal-8025 166 12 random random ADJ american_scientific_journal-8025 166 13 variables variable NOUN american_scientific_journal-8025 166 14 ” " PUNCT american_scientific_journal-8025 166 15 . . PUNCT american_scientific_journal-8025 167 1 mc mc PROPN american_scientific_journal-8025 167 2 tracts tracts PROPN american_scientific_journal-8025 167 3 . . PUNCT american_scientific_journal-8025 168 1 1970 1970 NUM american_scientific_journal-8025 169 1 [ [ X american_scientific_journal-8025 169 2 11 11 NUM american_scientific_journal-8025 169 3 ] ] X american_scientific_journal-8025 169 4 zwet zwet PROPN american_scientific_journal-8025 169 5 , , PUNCT american_scientific_journal-8025 169 6 w. w. PROPN american_scientific_journal-8025 169 7 β€œ " PUNCT american_scientific_journal-8025 169 8 mean mean VERB american_scientific_journal-8025 169 9 , , PUNCT american_scientific_journal-8025 169 10 median median ADJ american_scientific_journal-8025 169 11 , , PUNCT american_scientific_journal-8025 169 12 mode mode PROPN american_scientific_journal-8025 169 13 ii ii PROPN american_scientific_journal-8025 169 14 ” " PUNCT american_scientific_journal-8025 169 15 . . PUNCT american_scientific_journal-8025 170 1 statistica statistica PROPN american_scientific_journal-8025 170 2 neerlandica neerlandica VERB american_scientific_journal-8025 170 3 33 33 NUM american_scientific_journal-8025 170 4 , , PUNCT american_scientific_journal-8025 170 5 1 1 NUM american_scientific_journal-8025 170 6 – – SYM american_scientific_journal-8025 170 7 5 5 NUM american_scientific_journal-8025 170 8 . . NOUN american_scientific_journal-8025 170 9 1979 1979 NUM