^{1}

^{1}

In this paper, we use the generalized hypergeometric series method the high-order inverse moments and high-order inverse factorial moments of the generalized geometric distribution, the Katz distribution, the Lagrangian Katz distribution, generalized Polya-Eggenberger distribution of the first kind and so on.

The moment is one of the most widely used features of probability of random variables. The moments of random variables have been widely used in many important fields such as finance, probability theory, statistics and so on. So the calculation of the moment is very important. The inverse moment is a hot research direction in recent years. Inverse moment plays an important role in risk assessment, insurance and finance, and it is an important concept in probability. The study of the inverse moments originates from random sampling,

number of observations

sampling problem of inverse moment are introduced. Generally, the distribution of

The research on inverse moments of the binomial distribution and the Poisson distribution has been a long history. In 1945, Frederick F. Stephan studied the inverse moments of first and second order of the binomial distribution (see [

ments defined as

At present, more and more scholars are interested in the study of inverse moment, and have a wealth of research results mainly binomial distribution, Poisson distribution, negative binomial distribution, logarithmic distribution (see [

In the next, we will give some definitions necessarily.

Definition 1: Suppose X is a generalized geometric random variable with parameters

where

Definition 2: Suppose X is a generalized Polya-Eggenberger of the first kind random variable with parameters

where

Definition 3: Suppose X is a Katz random variable with parameters

where

Definition 4: Suppose X is a Lagrangian Katz random variable with parameters

where

The definition of generalized hypergeometric series:

where

If

In this section, we use a generalized hypergeometric series to obtain the inverse moments of some discrete distributions.

Theorem 2.1: Suppose

where

Proof. By definition 1, then

Note: when

Theorem 2.2: Suppose

where

Proof. By definition 2, then

Note: when

Let

Let

Let

Let

Corollary 2.1: Suppose

where

Proof. Let

Note: when

Corollary 2.2: Suppose ^{th} order is given by

where

Proof. Let

Note: when

In this section, we use generalized hypergeometric series to obtain the inverse factorial moments of some discrete distributions.

Theorem 3.1: Suppose

where

Proof. By definition 1, then

Note: when

Theorem 3.2: Suppose

where

Proof. By definition 2, then

Note: when

Let

Let

Let

Let

Corollary 3.1: Suppose

where

Proof. Let

Note: when

Corollary 3.2: Suppose

where

Proof. Let

Note: when

Bao, H.Y. and Wuyungaowa (2017) Application of Hyper- geometric Series in the Inverse Moments of Special Discrete Distribution. Journal of Applied Mathematics and Physics, 5, 267- 275. https://doi.org/10.4236/jamp.2017.52024