Poisson Distribution

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A Poisson distribution is the distribution of the number of events in a fixed time interval, provided that the events occur at random, independently in time and at a constant rate.

The event rate, µ, is the number of events per unit time. When µ is large, the shape of a Poisson distribution is very similar to that of the standard normal distribution. The change in shape of a Poisson distribution with increasing n is very similar to the equivalent binomial distribution. Convergence of distributions in this way can be explained by the central limit theorem.

Consider a time interval divided into many sub-intervals of equal length such that the probability of an event in a sub-interval is small and the probability of more than one event is negligible. If the probability of an event in each sub-interval is the same as and independent of that probability for other sub-intervals then n sub-intervals can be thought of as n independent trials. This is why Poisson distributions are closely related to binomial distributions.

Both the mean and variance of a Poisson distribution are equal to µ. The probability of r events happening in unit time with an event rate of µ is:

The summation of this Poisson frequency function from zero to r will always be equal to one as:

Analysis of mortality statistics often employs Poisson distributions on the assumption that deaths from most diseases occur independently and at random in populations (see Poisson rate confidence interval). Other common uses of Poisson are in Physics to model radioactive particle emission and in insurance companies to model accident rates.

Technical Validation

StatsDirect calculates cumulative probabilities that (≤,≥,=) r random events are contained in an interval when the average number of such events per interval is μ. The gamma function is a generalised factorial function and it is used to calculate each Poisson probability (Knusel, 1986). The core algorithm evaluates the logarithm of the gamma function (Cody and Hillstrom, 1967; Abramowitz and Stegun 1972; Macleod, 1989) to the limit of 64-bit precision. The inverse is found using a bisection algorithm to one order of magnitude larger than the limit of 64-bit precision.

Γ(*) is the gamma function:

Γ(1)=1

Γ(x+1)=xΓ(x)

Γ(n)=(n-1)!