A general type of statistical Distribution which is related to the Beta Distribution and arises naturally in processes for which the waiting times between Poisson Distributed events are relevant. Gamma distributions have two free parameters, labeled and , a few of which are illustrated above.

Given a Poisson Distribution with a rate of change , the Distribution Function giving the
waiting times until the th change is

(1) |

for . The probability function is then obtained by differentiating ,

(2) |

Now let and define to be the time between changes. Then the above equation can be written

(3) |

(4) |

(5) |

In order to find the Moments of the distribution, let

(6) | |||

(7) |

so

(8) |

and the logarithmic Moment-Generating function is

(9) | |||

(10) | |||

(11) |

The Mean, Variance, Skewness, and Kurtosis are then

(12) | |||

(13) | |||

(14) | |||

(15) |

The gamma distribution is closely related to other statistical distributions.
If , , ..., are independent random variates with a gamma distribution having parameters
,
, ...,
, then
is distributed as gamma with
parameters

(16) | |||

(17) |

Also, if and are independent random variates with a gamma distribution having parameters and , then is a Beta Distribution variate with parameters . Both can be derived as follows.

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

The sum therefore has the distribution

(24) |

(25) |

where is the Beta Function, which is a Beta Distribution.

If and are gamma variates with parameters and , the is a variate with a Beta
Prime Distribution with parameters and . Let

(26) |

(27) |

(28) |

(29) |

The ratio therefore has the distribution

(30) |

The ``standard form'' of the gamma distribution is given by letting
, so and

(31) |

so the Moments about 0 are

(32) |

(33) | |||

(34) | |||

(35) | |||

(36) |

The Moment-Generating Function is

(37) |

(38) |

(39) |

(40) |

**References**

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

© 1996-9

1999-05-25