Displaced Poisson distribution

In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution.

Definitions

Probability mass function

The probability mass function is

where and r is a new parameter; the Poisson distribution is recovered at r = 0. Here is the Pearson's incomplete gamma function:

where s is the integral part of r. The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution (that is ) is given by for and the displaced Poisson generalizes this ratio to .

Examples

One of the limitations of the Poisson distribution is that it assumes equidispersion – the mean and variance of the variable are equal. The displaced Poisson distribution may be useful to model underdispersed or overdispersed data, such as:

• the distribution of insect populations in crop fields;
• the number of flowers on plants;
• motor vehicle crash counts; and
• word or sentence lengths in writing.

Properties

Descriptive Statistics

• For a displaced Poisson-distributed random variable, the mean is equal to and the variance is equal to .
• The mode of a displaced Poisson-distributed random variable are the integer values bounded by and when . When , there is a single mode at .
• The first cumulant is equal to and all subsequent cumulants are equal to .

References