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

where and, ,.

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 and the pgf. Then, the distribution of X is zero-inflated Poisson if and only if, where, b are constants and is the first derivative of.

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 satisfies the linear differential equation

2) Suppose that the pgf of X satisfies

If, then and in turn. By the property of the pgf,. But, which is not possible because.

Therefore.

3) The Linear Differential Equation

The linear differential equation is of the form

where and are functions of.

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, using the above solution.

Since is a pgf, , where is the k-th derivative of.

We get

, , , and so on.

Now,

Since, it is easy to see that,

We have

with and.

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