| 释义 |
Poisson DistributionA 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) |
This should be normalized so that the sum of probabilities equals 1. Indeed,
 | (3) |
 | (4) |
so
The Moments about zero can also be computed directly
as can the Moments about the Mean.
so the Mean, Variance, Skewness, and Kurtosis are
The Characteristic Function is
 | (24) |
 | (25) |
 | (26) |
The Poisson distribution can also be expressed in terms of
 | (27) |
 | (28) |
 | (29) |
 ,  , ...,  have Poisson distributions with parameters  ,  ,...,  , then
 | (30) |
 | (31) |
 | (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. |
