Abstract
The purpose of the present paper is to introduce a generalized discrete probability distribution and obtain some results regarding moments, mean, variance, and moment generating function for this distribution. Further, we show that for specific values it reduces to various well-known distributions. Finally, we give a beautiful application of this distribution on certain analytic univalent functions.
Highlights
Let the series ∑∞ n=0 an, where an ≥ 0, ∀n ∈ N is convergent and its sum is denoted by S, that is, ∞ S = ∑an. (1) n=0we introduce the generalized discrete probability distribution whose probability mass function is p (n) = an S, n = 0, 1, 2, . . . . (2)Obviously p(n) is a probability mass function because p(n) ≥
The purpose of the present paper is to introduce a generalized discrete probability distribution and obtain some results regarding moments, mean, variance, and moment generating function for this distribution
We introduce the generalized discrete probability distribution whose probability mass function is p (n)
Summary
Let the series ∑∞ n=0 an, where an ≥ 0, ∀n ∈ N is convergent and its sum is denoted by S, that is,. We introduce the generalized discrete probability distribution whose probability mass function is p (n). From (1) it is easy to see that the series given by (3) is convergent for |x| < 1 and for x = 1 it is convergent. If X is a discrete random variable which can take the values x1, x2, x3, . The rth moment of a discrete probability distribution about X = 0 is defined by μr = E (Xr). The variance of the distribution is given by Variance = μ2 − (μ1 ). The moment generating function (m.g.f.) of a random variable X is denoted by MX(t) and defined by MX (t) = E (etX). The moment generating function of generalized discrete probability distribution is given by φ (et) MX (t) = S
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