Given a Poisson Distribution with rate of change , the distribution of waiting times between successive
changes (with ) is

(1) | |||

(2) |

which is normalized since

(3) |

This is the only Memoryless Random Distribution. Define the Mean waiting time between successive changes as . Then

(4) |

(5) | |||

(6) | |||

(7) |

so

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) |

The Skewness and Kurtosis are given by

(13) | |||

(14) |

The Mean and Variance can also be computed directly

(15) |

(16) |

(17) |

Now, to find

(18) |

(19) |

(20) |

giving

(21) | |||

(22) |

If a generalized exponential probability function is defined by

(23) |

(24) |

(25) | |||

(26) | |||

(27) | |||

(28) |

**References**

Balakrishnan, N. and Basu, A. P. *The Exponential Distribution: Theory, Methods, and Applications*.
New York: Gordon and Breach, 1996.

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 534-535, 1987.

Spiegel, M. R. *Theory and Problems of Probability and Statistics.* New York: McGraw-Hill, p. 119, 1992.

© 1996-9

1999-05-25