A distribution is a Gamma Distribution with
and
, where is the
number of Degrees of Freedom. If have Normal
Independent distributions with Mean 0 and Variance 1, then

(1) |

(2) |

(3) |

(4) |

where is a Regularized Gamma Function. The Confidence Intervals can be found by finding the value of for which equals a given value. The Moment-Generating Function of the distribution is

(5) | |||

(6) | |||

(7) | |||

(8) |

so

(9) | |||

(10) | |||

(11) | |||

(12) |

The th Moment about zero for a distribution with Degrees of Freedom is

(13) |

(14) | |||

(15) | |||

(16) |

The th Cumulant is

(17) |

The Moment-Generating Function is

(18) |

As ,

(19) |

(20) |

(21) |

(22) |

In a Gaussian Distribution,

(23) |

(24) |

(25) |

(26) |

(27) |

(28) |

(29) |

If are independent variates with a Normal Distribution having Means and
Variances for , ..., , then

(30) |

(31) |

The noncentral chi-squared distribution is given by

(32) |

(33) |

(34) | |||

(35) | |||

(36) | |||

(37) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 940-943, 1972.

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, p. 535, 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, pp. 115-116, 1992.

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1999-05-26