Application of Hypergeometric Series in the Inverse Moments of Special Discrete Distribution*

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1. Introduction

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, is the

number of observations with mean if

is independent and identically distributed random variable, the variance is,when is a constant, the variance of is, but when the is a random variable, the variance of was, at this point in the

sampling problem of inverse moment are introduced. Generally, the distribution of is mainly the Poisson distribution, binomial distribution and so on.

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 [1]). Grab and Stephan calculated tables of reciprocals for binomial and Poisson distribution as well as derive a recurrence relation. They also derived an exact expression for the first inverse moment (see [2]). Govindarajulu in 1963 a recursive formula moments of binomial distribution has been obtained (see [3]). In 1972, Chao and Strawderman (see [4]) considered slightly different inverse mo-

ments defined as which are frequently easier to calculate.

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 [5]). In this paper describes the use of generalized hypergeometric series inverse moments and factorial inverse moment distribution of some. It mainly includes Janardan discussed the distribution of the generalized Polya-Eggenberger distribution of the first kind, and the special value of the parameters (see [6]).

In the next, we will give some definitions necessarily.

Definition 1: Suppose X is a generalized geometric random variable with parameters, having probability mass function

(1)

where

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

(2)

where

Definition 3: Suppose X is a Katz random variable with parameters having probability mass function

(3)

where

Definition 4: Suppose X is a Lagrangian Katz random variable with parameters having probability mass function

(4)

where

The definition of generalized hypergeometric series:

where

If then is hypergeometric series.

2. The Inverse Moments of Some Discrete Distributions

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

Theorem 2.1: Suppose is a generalized geometric random variable with parameters, for, then the inverse moment of order is given by

where.

Proof. By definition 1, then

Note: when, the inverse moment of first order is given by

Theorem 2.2: Suppose is a generalized Polya-Eggenberger of the first kind random variable with parameters for then we have the inverse moment of order is given by

where

Proof. By definition 2, then

Note: when, the inverse moment of first order is given by

Let in theorem 2.2, then inverse moment of first order of the Polya-Eggenberger distribution is

Let in theorem 2.2, then inverse moment of first order of the binomial distribution is

Let in theorem 2.2, then can get the theorem 1 in the [5]

Let in theorem 2.2, then inverse moment of first order of the generalized Possion distribution is

Corollary 2.1: Suppose is a Katz random variable with parameters for then the inverse moment of order is given by

where

Proof. Let in theorem 2.2, By definition 3, then

Note: when, the inverse moment of first order is given by

Corollary 2.2: Suppose is a Lagrangian Katz random variable with parameters for, then the inverse moment of k^{th} order is given by

where

Proof. Let in theorem 2.2, by definition 4, then

Note: when, the inverse moment of first order is given by

3. The Factorial Inverse Moments of Some Discrete Distributions

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

Theorem 3.1: Suppose is a generalized geometric random variable with parameters, for, then the factorial inverse moment of order is given by

where

Proof. By definition 1, then

Note: when, the factorial inverse moment of first order is given by

Theorem 3.2: Suppose is a generalized Polya-Eggenberger of the first kind random variable with parameters for then we have the factorial inverse moment of order is given by

where

Proof. By definition 2, then

Note: when, the factorial inverse moment of first order is

Let in theorem 3.2, then factorial inverse moment of first order of the Polya-Eggenberger distribution is

Let in theorem 3.2, then factorial inverse moment of first order of the binomial distribution is

Let in theorem 3,.2, then can get the theorem 6 in the [5]

Let in theorem 3.2, then factorial inverse moment of first order of the generalized Possion distribution is

Corollary 3.1: Suppose is a Katz random variable with parameters for then the factorial inverse moment of order is given by

where

Proof. Let in theorem 3.2, by definition 3, then

Note: when, the factorial inverse moment of first order is given by

Corollary 3.2: Suppose is a Lagrangian Katz random variable with parameters for, then the factorial inverse moment of order is given by

where

Proof. Let in theorem 3.2, by definition 4, then

Note: when, the inverse factorial moment of first order is given by

NOTES

*Funded projects: The research is supported by the National Natural Science Foundation of China under Grant.

References

[1] Stephan, F.F. (1945) The Expected Value and Variance of the Reciprocal and Other Negative Powers of a Positive Bernoulli Variate. Annals of Mathematical Statistics, 16, 50-61. https://doi.org/10.1214/aoms/1177731170

[2] Grab, E.L. and Savage, I.R. (1954) Tables of the Expected of 1/x for Positive Bernoulli and Poisson Variable. Journal of American Statistical Association, 49, 167-177.

[3] Govindarajulu, Z. Recurrence Relation for the Inverse Moments of the Positive Binomial Variable. Journal of American Statistical Association, 58, 468-473.
https://doi.org/10.1080/01621459.1963.10500859

[4] Chao, M.T. and Strawderman, W.E. (1972) Negative Moments of Positive Random Variables. Journal of the American Statistical Association, 67, 429-431.
https://doi.org/10.1080/01621459.1972.10482404

[5] Wang, C.Y. and Wuyungaowa (2015) Recurrence Relation and Accurate Value on Inverse Mo-ment of Discrete Distributions. Journal of Probability and Statistics.
https://doi.org/10.1155/2015/972035

[6] Znidaric, M. (2005) Asymptotic Expansion for Inverse Moments of Binomial and Poisson Distribution. Statistics Theory [math.ST].