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 iswhere 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 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.