Given a Random Variable , if there exists an such that

(1) |

(2) |

(3) |

where is the th Moment about zero. The moment-generating function satisfies

(4) |

If is differentiable at zero, then the th Moments about the Origin are given by

(5) |

(6) |

(7) |

(8) |

(9) | |||

(10) |

It is also true that

(11) |

It is sometimes simpler to work with the Logarithm of the moment-generating function, which is also called the
Cumulant-Generating Function, and is defined by

(12) | |||

(13) | |||

(14) |

But , so

(15) | |||

(16) |

**References**

Kenney, J. F. and Keeping, E. S. ``Moment-Generating and Characteristic Functions,'' ``Some Examples of Moment-Generating Functions,''
and ``Uniqueness Theorem for Characteristic Functions.'' §4.6-4.8 in *Mathematics of Statistics, Pt. 2, 2nd ed.*
Princeton, NJ: Van Nostrand, pp. 72-77, 1951.

© 1996-9

1999-05-26