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.

(1) |

(2) |

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

(3) |

(4) |

(5) | |||

(6) | |||

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(8) | |||

(9) | |||

(10) |

so

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(12) |

The Moments about zero can also be computed directly

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(15) |

as can the Moments about the Mean.

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so the Mean, Variance, Skewness, and Kurtosis are

(20) | |||

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(22) | |||

(23) |

The Characteristic Function is

(24) |

(25) |

(26) |

The Poisson distribution can also be expressed in terms of

(27) |

(28) |

(29) |

(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.

© 1996-9

1999-05-25