Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 544 https://internationalpubls.com Zero Truncated Premium Linear-Exponential Mixture of Poisson Distribution 1Binod Kumar Sah, 2*Suresh Kumar Sahani 1Department of Statistics, R.R.M. Campus, JanakpurDham, Tribhuvan University, Nepal. Email: 1sah.binod01@gmail.com 2*Faculty of Science, Technology, and Engineering, Rajarshi Janak University, Janakpur Dham, Nepal. Email: 2*sureshsahani@rju.edu.np *Corresponding Author: sureshsahani@rju.edu.np Article History: Received: 12-10-2024 Revised: 15-11-2024 Accepted: 11-01-2025 Abstract: This distribution is a modified form of the Zero Truncated Poisson-Lindley distribution (ZTPLD), which has been shown to be better alternative of ZTPLD for statistical modeling of count data related to mortality and biological sciences. All the characteristics required for this distribution are presented and explained in very well manner. Keywords: Zero-truncated Probability distribution, Premium Linear-Exponential Mixture of Poisson distribution (PLEMPD), Premium Linear-exponential distribution (PLED), Chi-square Goodness of Fit Test, Probability distribution, Distribution. INTRODUCTION: Research is a regular process that allow us to constantly expand our knowledge by discovering new and unfamiliar things. The zero-truncated probability distribution is generated from a standard distribution excluding mass of the probability at zero. The proposed distribution is named as “Zero Truncated Premium Linear-Exponential Mixture of Poisson Distribution”. It is abbreviated as ZTPLEMPD. It is a compound discrete distribution. It may be called Conditional Premium Linear-exponential Mixture of Poisson Distribution (CPLEMPD) or Positive Premium Linear-exponential Mixture of Poisson Distribution (PPLEMPD) or Zero- truncated Poisson-Premium Linear-exponential Distribution (ZTPPLED), (see [1] to [4]). It is a left truncated probability distribution which is truncated at zero. This distribution an extension of Zero-truncated Poisson-Lindley distribution (ZTPLD) [5], which was obtained by using the definition of zero-truncated probability distribution as well as originated by size-biased Poisson distribution (1) when its parameter  follows a distribution having probability density function 1 ( / ) ; 1,2,3,...; 0 ( 1)! we g w w w    − − = =  − (1) given in the expression (2) as mailto:1sah.binod01@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 545 https://internationalpubls.com   2 0 2 ( , ) ( 1) ( 2) ( 3 1) h e          −= + + + + + (2) Hence, probability mass function of ZTPLD was obtained as 2 1 2 (2 ) ( , ) ( 3 1) (1 )w w P w       + + = + + + (3) Where 0  and 1,2,...,w=  For details account, see [6-15]. Works of this paper have been arranged under the following headings. 1.0 Introduction 2.0 Results 3.0 Applications 4.0 Conclusions 2.0 Results: 2.1 Probability Mass Function of ZTPLEMPD: It can be obtained by using (A) Definition of zero-truncated probability distribution (B) Mixing the expression (1) with (8). (A) ZTPLEMPD of variable w with parameter  is denoted by ( , )P w  and defined as 2 2 ( , ) ( , ) 1 ( 0, ) P w P w P w    = − = (4) Where 2 ( , )P w  is the Probability mass function of Premium Linear-exponential Mixture of Poisson Distribution (PLEMPD) [6] given by the equation (7).  2 2 2 2 1 (1 ) ( , ) (1 ) (1 )w w P w      + + + +  =   + +  (5) Where 0  and 0,1,2,3,...,w=  , and  2 2 2 2 1 (1 ) ( 0, ) (1 ) (1 ) P w      + +  = =   + +  (6) Putting the value of 2 ( , )P w  and 2 ( 0, )P w = in the expression (4) we get 2 2 3 2 (1 ) ( , ) ( 2 1) (1 )w w P w         + + + = + + + + (7) Where 0  and 1,2,...,w=  Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 546 https://internationalpubls.com (B) ZTPLEMPD can also be obtained by mixing (1) with (8) when its parameter  follows a distribution having probability density function given in the expression (8)   2 2 0 3 2 ( , ) ( 1) ( 1) ( 2 1) h e            −= + + + + + + + (8) Ans it can be derived as  2 1 (1 ) 0 ( , ) (1 ) ( 1)w wP w e d         − − + = + + + +   Or, 2 2 3 2 1 1 (1 ) ( 1) (1 ) ( ) ( , ) ( 1)!( 2 1) (1 ) (1 )w w w w P w w          +  +  + + +  = +  −+ + + + +  Or, 2 2 3 2 (1 ) ( , ) ( 2 1) (1 )w w P w          + + + =   + + + +  (9) Where 0  and 1,2,...,w=  . The expression (9) is the pmf of ZTPLEMPD. Figure-1: Showing graph of pmf of ZTPLEMPD Figure-2: Showing graph of pmf of ZTPLEMPD Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 547 https://internationalpubls.com Figure-3: Showing graph of pmf of ZTPLEMPD 2.2 Factorial Moments ( ( )r ), Moments about Origin ( r ) and Central Moments ( r ) of ZTPLEMPD: The factorial moment of order r and the first four factorial moments, the first four moments about the origin and the first four moments about the mean have been derived and given by the expression (10) to (22) in order respectively.  ( ) /r r E w  = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 548 https://internationalpubls.com Or,   2 ( ) 1 2 ( ) 2 3 10 ( 1) ( 1 ( 1)!(1 2 ) r w r w w e w e d w             − − − =    = + + + +  −+ + +    Or,   2 1 2 ( ) 2 3 0 ( ) ( 1) ( 1 (1 2 ) r r r w e d            − −  = + + + + + + + +  Or, 2 2 ( ) 2 3 (1 ) ! (1 ) (1 2 ) r r r r        +  = + + + + + (10) Therefore, from the expression (11) to (14) are first four factorial moments of ZTPLEMPD in order respectively. 2 2 (1) 2 3 (1 ) 1 (2 ) (1 2 )       +  = + + + + (11) 2 2 (2) 2 3 2 (1 ) 2 (3 ) (1 2 )        +  = + + + + (12) 2 2 (3) 2 3 3 (1 ) 6 (4 ) (1 2 )        +  = + + + + (13) 2 2 (4) 2 3 4 (1 ) 24 (5 ) (1 2 )        +  = + + + + (14) Conversion of Factorial Moments about Origin into Moments about the Origin: 2 2 1 (1) 2 3 (1 ) (2 ) (1 2 )         + +  = = + + + (15) 2 2 2 2 2 (2) (1) 2 2 3 2 3 2(1 ) (3 ) (1 ) (2 ) (1 2 ) (1 2 )                + + + +   = + = + + + + + + + Or, 2 2 3 2 2 2 3 (1 ) (6 2 2 ) (1 2 )          + + + +  = + + + (16) 2 2 2 2 2 2 3 (3) (2) (1) 3 2 3 2 2 3 2 3 6(1 ) (4 ) 6(1 ) (3 ) (1 ) (2 ) 3 (1 2 ) (1 2 ) (1 2 )                       + + + + + +    = + + = + + + + + + + + + + + 2 2 2 3 3 2 2 3 (1 ) (24 18 2 6 6 ) (1 2 )           + + + + +  = + + + (17) 2 2 2 2 4 (4) (3) (2) (1) 4 2 3 3 2 3 2 2 2 2 2 2 3 2 3 24(1 ) (5 ) 36(1 ) (4 ) 6 7 (1 2 ) (1 2 ) 14(1 ) (3 ) (1 ) (2 ) (1 2 ) (1 2 )                              + + + +     = + + + = + + + + + + + + + + + + + + + + + + + Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 549 https://internationalpubls.com Or, 2 2 2 3 3 4 5 4 4 2 3 (1 ) (120 144 42 24 2 36 14 ) (1 2 )              + + + + + + + +  = + + + (18) The first four moments about the mean has been obtained and expressed by the equations from (19) to (22) respectively. 1 0 = (19) 2 2 3 2 2 2 2 2 2 3 2 3 (1 ) ( 2 2 6) (1 ) (2 2) (1 2 ) (1 2 )                 + + + + + + = −   + + + + + +  0r, 2 2 5 2 4 3 2 2 2 2 3 2 (1 ) ( 5 4 6 2) (1 2 )              + + + + + + = + + + (20) 2 2 2 3 4 2 2 3 3 3 2 3 2 2 3 3 2 2 2 2 2 3 2 3 (1 ) (24 18 2 6 6 ) (1 ) (6 2 2 ) 3 (1 2 ) (1 2 ) (1 ) (2 ) (1 ) (2 ) 2 (1 2 ) (1 2 )                                 + + + + + + + + + + = −   + + + + + +     + + + + +    + + + + + +    2 2 2 3 4 2 3 2 4 2 2 3 2 3 6 2 3 3 3 2 3 3 (1 ) (24 18 2 6 6 ) (1 2 ) 3(1 ) (2 )(6 2 2 ) (1 2 ) 2(1 ) (2 ) (1 2 )                          + + + + + +   + + + − + + + + +   + + + + + +  = + + +          (21) 2 2 2 3 3 4 5 4 4 2 3 2 2 2 3 4 2 2 3 2 3 2 3 2 2 3 2 2 3 (1 ) (120 144 42 24 2 36 14 ) (1 2 ) (1 ) (24 18 2 6 6 ) (1 ) (2 ) 4 (1 2 ) (1 2 ) (1 ) (6 2 2 ) 6 (1 2 )                                      + + + + + + + + = + + +    + + + + + + + + −     + + + + + +     + + + + −  + + +  2 4 2 2 2 2 2 3 2 3 (1 ) (2 ) (1 ) (2 ) 3 (1 2 ) (1 2 )                + + + + +     + + + + + +    2 2 2 3 3 4 5 2 3 3 4 2 2 3 4 2 2 3 2 6 2 3 2 2 2 3 8 2 4 4 4 2 3 4 (1 ) (120 144 42 24 2 36 14 ) (1 2 ) 4(1 ) (24 18 2 6 6 ) (2 )(1 2 ) 6(1 ) (6 2 2 ) (2 ) (1 2 ) 3(1 ) (2 ) (1 2 )                                      + + + + + + + + + + + − + + + + + + + + + + + + + + + + + + + − + + = + + +                           (22) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 550 https://internationalpubls.com 2.3 Nature of ZTPLEMPD Based on Variability, Shape and Size: • Based on Variability: 2 5 2 4 3 2 2 2 3 ( 5 4 6 2) (2 )(1 2 ) I            + + + + + = + + + + (23) It is index of dispersion. For over-dispersion Or, 2 5 4 2 2 2 4 3 2( 4 4 4 2 4 4 2) 0         + + + − − − − −  (24) Table-1 Serial Number Types of Variation Condition 1 Over-dispersion If 1.14930004  2 Equi-dispersed If 1.14930004 = 3 Under-dispersed If 1.14930004  For details (see, [15]). • Coefficient of Skewness ( 1 ):   2 2 2 3 4 2 3 2 4 2 2 3 2 3 6 2 3 1 3/2 2 2 5 2 4 3 2 (1 ) (24 18 2 6 6 ) (1 2 ) 3(1 ) (2 )(6 2 2 ) (1 2 ) 2(1 ) (2 ) (1 ) ( 5 4 6 2)                              + + + + + +   + + + − + + + + +    + + + + + +  = + + + + + + (24) • Coefficien of Kurtosis ( 2 ): 2 2 2 3 3 4 5 2 3 3 4 2 2 3 4 2 2 3 2 6 2 3 2 2 2 3 8 2 4 2 (1 ) (120 144 42 24 2 36 14 ) (1 2 ) 4(1 ) (24 18 2 6 6 ) (2 )(1 2 ) 6(1 ) (6 2 2 ) (2 ) (1 2 ) 3(1 ) (2 ) (1                                   + + + + + + + +   + + + − + + + + + +    + + + + + + + + +   + + + + − + +  = +  2 2 2 5 2 4 3 2) ( 5 4 6 2)      + + + + + (25) Remarks: • 1( 2)    • 26    • Hence the proposed distribution is positively skewed and leptokurtic in nature. 2.4 Estimation of Parameter: It can be obtained by using the following equation which has been derived by using 1 of this distribution. 4 3 2( 1) (2 ) ( 3) 2 0k k k k     + − + − + − − = (26) Where 1k w= − Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 551 https://internationalpubls.com 3.0 Applications of ZTPLEMPD: We may use this distribution for modelling of zero-truncated count data related to Mortality as well as Social Sciences (see, [16] to [21]). The example related to mortality are from (1) to (8) and the examples (9) and (10) are related to social sciences which are given in table from (2) to (11) respectively. Ultimately, we have used ten examples on which have applied chi-square goodness of fit test by using ZTPD as well as ZTPLEMPD which are placed in the table numbered from (12) to (21) in order respectively which indicate superiority of ZTPLEMPD over ZTPD. Example-1 Table-2 The table shows one live birth and at least one neonatal death of mothers of rural area, where W represents the number of neonatal death and observed number of mothers W 1 2 3 4 5 O 409 88 19 5 1 Example-2 Table-3 The table shows the no. of mothers from estate area who have experienced one live birth and at least one neonatal death, where W represents the number of neonatal death and observed number of mothers. W 1 2 3 4 5 O 71 32 7 5 3 Example-3 Table-4 The table shows the number of mothers from urban area who have experienced at least two live births by the numbers of infant and child deaths W 1 2 3 4 5 O 176 44 16 6 2 Example-4 Table-5 The table shows the number of mothers from rural area who have experienced at least two live births by the numbers of infant and child deaths W 1 2 3 4 5 6 O 745 212 50 21 7 2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 552 https://internationalpubls.com Example-5 Table-6 The table shows the number of literate mothers who have experienced of at least one live birth and at least one death W 1 2 3 4 5 O 683 145 29 11 5 Exaple-6 Table-7 The table shows the number of mothers who have completed fertility with at least one child death. W 1 2 3 4 5 6 O 89 25 11 6 3 1 Exaple-7 Table-8 The table shows the number of mothers who have experienced at least one infant death W 1 2 3 4 5 O 567 135 28 11 5 Exaple-8 Table-9 The table shows the no. of European res mites on apple leaves reported by Garman [19]. W 1 2 3 4 5 6 7 8 O 38 17 10 9 3 2 1 0 Where W and O represents the no. red mites and the no. of observed leaves respectively. Exaple-9 Table-10 The table shows the count of yeast cell per mm square reported by Students [20]. W 1 2 3 4 5 6 O 128 37 18 3 1 0 Where W and O represents the no. of Yeast cell per mm square and the no. of observed leaves respectively. Exaple-10 Table-11 The table shows the count of snows hares counts captured over 7 days reported by Keith and Meslow [21]. W 1 2 3 4 5 O 184 55 14 4 4 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 553 https://internationalpubls.com Where W and O represents the no. of snows hares caught and the no. of observed respectively. Table-12 Fitting of ZTPLD and ZTPLEMPD to the example (1) W O Theoretical frequency due to ZTPLD ZTPLEMPD 1 409 408.1 408.5 2 88 89.4 88.8 3 19 19.3 19.3 4 5 4.1 4.2 5 1 1.1 1.2 Total 522 522.0 522.0 1.277777w= 1 0.277777m w= − = ̂ 4.199697 3.68115 . .d f 2 2 2 0.145 0.079 P value− 0.9301 0.9613 Table-13 Fitting of ZTPLD and ZTPLEMPD to the example (2) W O Theoretical frequency due to ZTPLD ZTPLEMPD 1 71 72.3 72.8 2 32 28.4 27.9 3 7 10.9 10.6 4 5 4.1 4.1 5 3 2.3 2.6 Total 118 118.0 118.0 1.618644068w= 1 0.618644068m w= − = ̂ 2.0496094 1.75878 . .d f 2 2 2 2.274 2.121 P value− 0.3204 0.3463 Table-14 Fitting of ZTPLD and ZTPLEMPD to the example (3) W O Theoretical frequency due to ZTPLD ZTPLEMPD Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 554 https://internationalpubls.com 1 176 171.6 172.0 2 44 53.3 50.8 3 16 15.0 15.0 4 6 4.3 4.4 5 2 1.7 1.8 Total 244 244.0 244.0 1.418032787w= 1 0.418032787m w= − = ̂ 2.209411 2.50388 . .d f 2 2 2 1.882 1.5915 P value− 0.3902 0.4516 Table-15 Fitting of ZTPLD and ZTPLEMPD to the example (4) W O Theoretical frequency due to ZTPLD ZTPLEMPD 1 745 738.1 739.9 2 212 214.8 212.6 3 50 61.3 61.0 4 21 17.2 17.5 5 7 4.8 5.0 6 3 1.8 2.0 Total 1038 1038.0 1038.0 1.402697495w= 1 0.402697495m w= − = ̂ 3.007722 2.59203 . .d f 3 3 2 4.773 4.0061 P value− 0.1892 0.2608 Table-16 Fitting of ZTPLD and ZTPLEMPD to the example (5) W O Theoretical frequency due to ZTPLD ZTPLEMPD 1 683 674.4 675.0 2 145 154.1 153.1 3 29 34.6 34.7 4 11 7.7 7.9 5 5 2.2 2.3 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 555 https://internationalpubls.com Total 873 873.0 873.0 1.293241695w= 1 0.293241695m w= − = ̂ 4.00231 3.49508 . .d f 2 2 2 5.310 4.7577 P value− 0.0703 0.0927 Table-17 Fitting of ZTPLD and ZTPLEMPD to the example (6) W O Theoretical frequency due to ZTPLD ZTPLEMPD 1 89 83.4 83.9 2 25 32.3 31.8 3 11 12.2 12.0 4 6 4.5 4.5 5 3 1.6 1.7 6 1 0.9 1.1 Total 135 135.0 135.0 1.607407407w= 1 0.607407407m w= − = ̂ 2.089084 1.78732 . .d f 2 2 2 3.428 2.8460 P value− 0.1801 0.2410 Table-18 Fitting of ZTPLD and ZTPLEMPD to the example (7) W O Theoretical frequency due to ZTPLD ZTPLEMPD 1 567 561.4 562.1 2 135 139.7 138.6 3 28 34.2 34.2 4 11 8.2 8.4 5 5 2.6 2.7 Total 746 746.0 746.0 1.327077748w= 1 0.327077748m w= − = ̂ 3.625737 3.15020 . .d f 2 2 2 3.839 3.4232 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 556 https://internationalpubls.com P value− 0.1467 0.1808 Table-19 Fitting of ZTPLD and ZTPLEMPD to the example (8) W O Theoretical frequency due to ZTPLD ZTPLEMPD 1 38 36.1 36.9 2 17 20.5 20.1 3 10 11.5 10.8 4 9 5.6 5.8 5 3 3.1 3.1 6 2 1.6 1.6 7 1 0.8 0.9 8 0 0.8 0.8 Total 80 80.0 80.0 2.15w= 1 1.15m w= − = ̂ 1.185582 1.04542665 . .d f 3 3 2 2.467 2.3606 P value− 0.4813 0.5010 Table-20 Fitting of ZTPLD and ZTPLEMPD to the example (9) W O Theoretical frequency due to ZTPLD ZTPLEMPD 1 128 127.6 128.1 2 37 40.9 40.4 3 18 12.8 12.7 4 3 4.0 4.0 5 1 1.2 1.3 6 0 0.5 0.5 Total 187 187.0 187.0 1.459893048w= 1 0.459893048m w= − = ̂ 2.667323 2.29372882 . .d f 1 1 2 1.034 0.948 P value− 0.309 0.330 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 557 https://internationalpubls.com Table-21 Fitting of ZTPLD and ZTPLEMPD to the example (10) W O Theoretical frequency due to ZTPLD ZTPLEMPD 1 184 182.6 183.1 2 55 55.3 54.7 3 14 16.4 16.3 4 4 4.8 4.9 5 4 1.9 2.0 Total 261 261.0 261.0 1.425287356w= 1 0.425287356m w= − = ̂ 2.863957 2.46444202 . .d f 2 2 2 0.61 0.5059 P value− 0.7371 0.7765 4.0 Conclusion: • It is suggested to apply ZTPNLEMPD instead of ZTPLD for the zero-truncated count data related to mortality and social sciences because P-value obtained by using ZTPLEMPD is greater than those obtained by using ZTPLD for the table numbered from (12) to (21). • This distribution is Positively skewed and Leptokurtic in nature. • This distribution is over-dispersed if 1.14930004  . Conflict of Interest: The sole purpose of this paper is to contribute to the field of zero-truncated Poisson-Continuous distribution. Our intention is not to hurt anyone’s feelings. Acknowledgement: Heartfelt thanks to the editor-in-chief, editors, Production unit of this journal and everyone who directly or indirectly contributed to improving the quality of this paper. References: [1] Sah, B.K. (2022). Premium Linear-exponential Distribution. Applied Science Periodical,24 (3), 1-17. [2] Sah, BK and Sahani, SK (2024). Premium Linear-exponential Mixture of Poisson Distribution. Communication of Applied Nonlinear Analysis,31(1,187-199). [3] Sah, BK (2022). New Linear-Exponential Distribution. Applied Science Periodical, 24(2), 01-17. (4) Sankaran. M (1970). The Discrete Poisson-Lindley Distribution. Biometrics, 26, 145-149. 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