# moment generating function

Given a random variable^{} $X$, the *moment generating function* of $X$ is the following function:

${M}_{X}(t)=E[{e}^{tX}]$ for $t\in R$ (if the expectation converges).

It can be shown that if the moment generating function of $X$ is defined on an interval around the origin, then

$E[{X}^{k}]={{M}_{X}^{(k)}(t)|}_{t=0}$

In other words, the $k$th-derivative of the moment generating function evaluated at zero is the $k$th moment of $X$.

Title | moment generating function |

Canonical name | MomentGeneratingFunction |

Date of creation | 2013-03-22 11:53:51 |

Last modified on | 2013-03-22 11:53:51 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 10 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 60E05 |

Classification | msc 46L05 |

Classification | msc 82-00 |

Classification | msc 83-00 |

Classification | msc 81-00 |

Related topic | CharacteristicFunction2 |

Related topic | CumulantGeneratingFunction |