Microsoft Word - Communications on Applied Nonlinear Analysis_pro Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 8s (2025) Memory Effect of Deformable Mean and Deformable Standard Deviation Hari Pratap1, Sunit Kumar2*, Subodh Kumar3, Gajraj Singh4, Ved Pal Singh5 1Department of Mathematics, P.G.D.A.V. College Evening, University of Delhi-110065 haripratap@pgdave.du.ac.in 2Department of Mathematics, Motilal Nehru College, University of Delhi-110021 sunit@mln.du.ac.in 3Department of Mathematics, Shyam Lal College, University of Delhi-110032 skumarmath@shyamlal.du.ac.in 4Discipline of Statistics, School of Sciences, Indira Gandhi National Open University, Delhi gajrajsingh@ignou.ac.in 5Department of Mathematics, Maharaja Agersen College, University of Delhi-110096 vedpalsingh@mac.du.ac.in *Corresponding author: Sunit Kumar Article History: Received: 30-10-2024 Revised: 04-12-2024 Accepted: 27-12-2024 Abstract: Memory effects can be easily understood with the help of fractional derivatives. In this paper with the help of Deformable fractional derivative we try to understand the role of deformability on memory effects of a normal distribution using moment generating function. First, we develop a formula for Deformable mean and Deformable standard deviation with the help of Deformable fractional derivative. Secondly, we calculate the value of Deformable mean and Deformable standard deviation for varying fractional order. Lastly, we draw plot of normal distribution for calculated value of Deformable mean and Deformable deviation with varying fractional order and give conclusion. Keywords: Moment generating function, Fractional order, Deformable derivative, Deformable mean and Deformable standard deviation. 1. Introduction On the basis of Deformable fractional derivative, we will define Deformable mean and Deformable standard deviation. Deformable Fractional Derivative: Let 𝑓(π‘₯) be a real valued function defined on interval (π‘Ž, 𝑏) for a given number 𝛼, 0 ≀ 𝛼 ≀ 1 lim β†’ ( ) ( ) ( ) (1) Where 𝛼 + 𝛽 = 1 If this limit exists, we denote it by 𝐷 [𝑓(π‘₯)] [1]. 𝐷 [𝑓(π‘₯)] = 𝛼 𝐷𝑓(π‘₯) + 𝛽𝑓(π‘₯) (2) or 𝐷 [𝑓(π‘₯)] = 𝛼 𝐷𝑓(π‘₯) + (1 βˆ’ 𝛼) 𝑓(π‘₯) https://internationalpubls.com 439 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 8s (2025) Moment Generating Function: Let 𝑋 be random variable (r.v.) such that for some πœ– > 0, the expected value of 𝑒 exists for βˆ’πœ– < 𝑑 < πœ–. Then moment generating function (m.g.f.) of 𝑋 is defined to be the function 𝑀 (𝑑) = 𝐸[𝑒 ], for βˆ’πœ– < 𝑑 < πœ– [3],[5]-[6]. 𝑀 (𝑑) = 𝐸[𝑒 ] = ∫ 𝑒 𝑓 (π‘₯)𝑑π‘₯ (3) Where 𝑋 is a continuous r.v. and 𝑓 (π‘₯) is a probability density function s.t. [3], [5]. (i) 𝑓 (π‘₯) β‰₯ 0 (ii) ∫ 𝑓 (π‘₯)𝑑π‘₯ = 1 or 𝑀 (𝑑) = 𝐸[𝑒 ] = 𝑒 𝑝 (π‘₯) ∈ Where 𝑋 is a discrete r.v. with space Ξ© and 𝑝 (π‘₯) is a probability mass function s.t. [3], [5]. (i) 0 ≀ 𝑝 (π‘₯) ≀ 1 , π‘₯ ∈ Ξ© (ii) βˆ‘ 𝑝 (π‘₯)∈ = 1 2. Deformable Mean If 𝑀 (𝑑) is a well-defined moment generating function then we can define Deformable mean (πœ‡ ) with the help of moment generating function as πœ‡ = 𝑀 (0) Deformable derivative of equation (3) w.r.to t 𝑀 (𝑑) = ∫ 𝐷 [𝑒 ] 𝑓 (π‘₯)𝑑π‘₯ (4) 𝑀 (𝑑) = {𝛼 π‘₯ 𝑒 + (1 βˆ’ 𝛼) 𝑒 } 𝑓 (π‘₯)𝑑π‘₯ 𝑀 (𝑑) = 𝛼 ∫ π‘₯ 𝑒 𝑓 (π‘₯) 𝑑π‘₯ + (1 βˆ’ 𝛼) ∫ 𝑒 𝑓 (π‘₯)𝑑π‘₯ (5) 𝑀 (0) = 𝛼 π‘₯ 𝑒 𝑓 (π‘₯) 𝑑π‘₯ + (1 βˆ’ 𝛼) 𝑒 𝑓 (π‘₯)𝑑π‘₯ 𝑀 (0) = 𝛼 π‘₯ 𝑓 (π‘₯) 𝑑π‘₯ + (1 βˆ’ 𝛼) 𝑓 (π‘₯) 𝑑π‘₯ 𝑀 (0) = 𝛼 𝐸(𝑋) + (1 βˆ’ 𝛼) (6) or πœ‡ = 𝛼 πœ‡ + (1 βˆ’ 𝛼) For πœ‡ (Deformable mean) following cases arises: (i) 𝛼 = 0, and πœ‡ ∈ 𝑅 then πœ‡ =1 (7) https://internationalpubls.com 440 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 8s (2025) (ii) 𝛼 = , and πœ‡ ∈ 𝑅 then πœ‡ = (πœ‡ + 1) (8) (iii) 𝛼 = 1, and πœ‡ ∈ 𝑅 then πœ‡ = πœ‡ (9) (iv) 0 < 𝛼 < 1, and πœ‡ ∈ 𝑅 then πœ‡ = π›Όπœ‡ + (1 βˆ’ 𝛼) (10) 3. Deformable Standard Deviation If 𝑀 (𝑑) is a well-defined moment generating function then we denote Deformable variance by (𝜎 ) and defined as (𝜎 ) = 𝑀 . (0) βˆ’ (𝑀 (0)) (11) Deformable derivative of equation (5) w.r.to t 𝑀 . (𝑑) = 𝛼 ∫ π‘₯ 𝐷 { 𝑒 } 𝑓 (π‘₯) 𝑑π‘₯ + (1 βˆ’ 𝛼) ∫ 𝐷 {𝑒 } 𝑓 (π‘₯)𝑑π‘₯ 𝑀 . (𝑑) = 𝛼 π‘₯ {𝛼 π‘₯ 𝑒 + (1 βˆ’ 𝛼) 𝑒 } 𝑓 (π‘₯) 𝑑π‘₯ + (1 βˆ’ 𝛼) {𝛼 π‘₯ 𝑒 + (1 βˆ’ 𝛼) 𝑒 } 𝑓 (π‘₯)𝑑π‘₯ 𝑀 . (𝑑) = 𝛼 π‘₯ 𝑒 𝑓 (π‘₯) 𝑑π‘₯ + 2 𝛼 (1 βˆ’ 𝛼) π‘₯ 𝑒 𝑓 (π‘₯)𝑑π‘₯ + (1 βˆ’ 𝛼) 𝑒 𝑓 (π‘₯)𝑑π‘₯ 𝑀 . (0) = 𝛼 π‘₯ 𝑒 𝑓 (π‘₯) 𝑑π‘₯ + 2 𝛼 (1 βˆ’ 𝛼) π‘₯ 𝑒 𝑓 (π‘₯)𝑑π‘₯ + (1 βˆ’ 𝛼) 𝑒 𝑓 (π‘₯)𝑑π‘₯ 𝑀 . (0) = 𝛼 𝐸(𝑋 ) + 2 𝛼 (1 βˆ’ 𝛼)𝐸(𝑋) + (1 βˆ’ 𝛼) (12) (𝜎 ) = 𝛼 𝐸(𝑋 ) + 2 𝛼 (1 βˆ’ 𝛼)𝐸(𝑋) + (1 βˆ’ 𝛼) βˆ’ 𝛼 𝐸(𝑋) βˆ’ (1 βˆ’ 𝛼) βˆ’ 2 𝛼 (1 βˆ’ 𝛼) 𝐸(𝑋) (𝜎 ) = 𝛼 𝜎 𝜎 = 𝛼 𝜎 (13) We get a linear relation between Deformable standard deviation and an ordinary standard deviation. Deformable standard deviation is 𝛼-times the ordinary standard deviation. For 𝛼 = 0, Deformable standard deviation is zero. As fractional order increases Deformable standard deviation also increase. 𝜎 ≀ 𝜎 (14) Example 1: For a continuous random variable 𝑋, Probability density function of normal distribution is defined as 𝑓 (π‘₯) = √ 𝑒 (15) Where βˆ’βˆž < π‘₯ < ∞ and, βˆ’βˆž < πœ‡ < ∞, 𝜎 > 0 [3], [5]. https://internationalpubls.com 441 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 8s (2025) On the basis of equations (6) and (13) calculated value of Deformable mean and Deformable standard deviation given in the following table, for mean πœ‡ = 50 and standard deviation 𝜎 = 25.3 and fractional order 𝛼 = 0.3 , 0.5 , 0.7 , 0.9 , 1. Fractional Order (𝜢) Deformable Mean ( 𝝁𝜢) Deformable S.D. (πˆπ‘Ώ 𝜢) 0.3 15.7 7.59 0.5 25.5 12.65 0.7 35.3 17.71 0.9 45.5 22.77 1 50 25.3 Figure: Normal distribution plot of fractional order 𝛼 = 0.3 , 0.5 , 0.7 , 0.9 , 1. 4. Conclusion Deformable and ordinary mean have a linear relationship and Deformable standard deviation is less than and equal to ordinary standard deviation. Therefore, Deformable standard deviation is more reliable as compare to ordinary standard deviation. Deformable mean along with Deformable standard deviation gives more insight information of a normal distribution. We can see that Deformable derivative base mean and standard deviation affects the sharpness and flatness of a given normal distribution, which shows the memory of a normal distribution over varying fractional order. Therefore, we can say that moment generating base Deformable means and Deformable standard deviation is extension or ordinary mean and standard deviation. Further we can also extend this result to other measure of dispersion. References 1. Fahed Zulfeqar, Amit Ujlayan, Priyanka Ahuja, A New Fractional Derivative and Its Fractional Integral with Some Applications, XIV: 1705. 00962v1 [math.CA], 2017. https://internationalpubls.com 442 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 8s (2025) 2. R. Khalil, M. AL Horan, A. Yousef, & M. Sababheh, A New Definitions of Fractional Derivative. Elsevier, Journal of Computational and Applied Mathematics, 2014, 264, 65-70. 3. Miller, Irvin & Miller, Marylee’s John E. Freund’s, Mathematical Statistics with Applications (7th edition). Pearson Education Asia, 2006. 4. Miller, Irvin & Miller, Marylee’s John E. Freund’s, Mathematical Statistics with Applications (7th edition). Pearson Education Asia, 2006. 5. Jay L. Devore, & Kenneth N. Berk, Mathematical Statistics with Applications, Thomson Brooks Cole, 2007. 6. Robert V. Hogg, Alen Craig, Joseph W. McKean, Introduction to Mathematical Statistics, Sixth Edition, Pearson, 2011. 7. S.P. Gupta, Statistical Method, Sultan Chand & Sons, 2013. 8. K.B. Oldham, & J. Spanier, The Fractional Calculus, New York, Academic Press, 1974. 9. Udita N. Katugampola, New Fractional Derivative with Classical Properties, Journal of The American Mathematical Society, vol. 00, No. 0, pp. 000-000, arXiv:1410.6535v2 [math.CA] 8 Nov. 2014. 10. Udita N. Katugampola, A new Fractional Derivative with Classical Properties, Journal of American Mathematical Society volume, 2014. 11. Tarasov, Vasily E., No Violation of the Leibniz Rule, No Fractional Derivative, Elsevier, Communication to Nonlinear Science and Numerical Simulation, 2013, 18, 2945-2948. 12. Tarasov, Vasily E., On Chain Rule of Fractional Derivatives, Elsevier, Communication to Nonlinear Science and Numerical Simulation, 30 (2016), pp. 1-4. 13. Abdeljawad, T., On Conformable Fractional Calculus, Elsevier, Journal of Computational and Applied Mathematics, 2015, 279, 57-66. https://internationalpubls.com 443