## Poisson Distribution

A Poisson distribution is a distribution with the following properties:

1. The number of changes in nonoverlapping intervals are independent for all intervals.

2. The probability of exactly one change in a sufficiently small interval is , where is the probability of one change and is the number of Trials.

3. The probability of two or more changes in a sufficiently small interval is essentially 0.

The probability of changes in a given interval is then given by the limit of the Binomial Distribution
 (1)

as the number of trials becomes very large,

 (2)

This should be normalized so that the sum of probabilities equals 1. Indeed,
 (3)

as required. The ratio of probabilities is given by
 (4)

The Moment-Generating Function of this distribution is
 (5) (6) (7) (8) (9) (10)

so
 (11) (12)

The Moments about zero can also be computed directly
 (13) (14) (15)

as can the Moments about the Mean.
 (16) (17) (18) (19)

so the Mean, Variance, Skewness, and Kurtosis are
 (20) (21) (22) (23)

 (24)

and the Cumulant-Generating Function is
 (25)

so
 (26)

The Poisson distribution can also be expressed in terms of

 (27)

the rate of changes, so that
 (28)

The Moment-Generating Function of a Poisson distribution in two variables is given by
 (29)

If the independent variables , , ..., have Poisson distributions with parameters , , ..., , then
 (30)

has a Poisson distribution with parameter
 (31)

This can be seen since the Cumulant-Generating Function is
 (32)

 (33)

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 532, 1987.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function.'' §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209-214, 1992.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 111-112, 1992.