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Zero-Inflated Poisson model has found a wide variety of applications in recent years in statistical analyses of count data, especially in count regression models. Zero-Inflated Poisson model is characterized in this paper through a linear differential equation satisfied by its probability generating function [1] [2].

A random variable X is said to have a zero-inflated Poisson distribution if its probability mass function is given by

where

Thus, the distribution of X is a mixture of a distribution degenerate at zero and a Poisson distribution with mean

The probability generating function (pgf) of X is given by

Let X be a non-negative integer valued random variable with

Proof:

1) Suppose that X has a zero-inflated Poisson distribution specified in (1.1). Then the pgf of X is given by

On differentiation, we get

Hence

2) Suppose that the pgf

If

Therefore

3) The Linear Differential Equation

The linear differential equation

where

Then its solution is given by

where c is an arbitrary constant.

Here

Hence

Therefore the solution of the Equation (2) is given by

We now extract the probabilities

Since

We get

Now,

Since

We have

with

Therefore X has the pgf specified in Equation (1).