1. Introduction
A random variable X is said to have a zero-inflated Poisson distribution if its probability mass function is given by
(1)
![](//html.scirp.org/file/6-1240489x6.png)
where
and![](//html.scirp.org/file/6-1240489x9.png)
,
,
.
Thus, the distribution of X is a mixture of a distribution degenerate at zero and a Poisson distribution with mean
.
2. Probability Generating Function
The probability generating function (pgf) of X is given by
![](//html.scirp.org/file/6-1240489x13.png)
.
3. Characterization
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)
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). ![]()