# Poisson random variable

The Poisson discrete probability function with parameter $\lambda >0$ is given by

$${f}_{X}(x)=\frac{{e}^{-\lambda}{\lambda}^{x}}{x!},x\in \mathbb{N}.$$ |

A random variable^{} $X$ with such a density has expectation, variance^{}, moment generating function and characteristic function^{} given by $E[X]=\lambda $, $Var[X]=\lambda $, ${M}_{X}(t)={e}^{\lambda ({e}^{t}-1)}$, and ${\varphi}_{X}(t)={e}^{\lambda ({e}^{it}-1)}$, respectively.

Title | Poisson random variable |
---|---|

Canonical name | PoissonRandomVariable |

Date of creation | 2013-03-22 11:54:03 |

Last modified on | 2013-03-22 11:54:03 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 13 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 62E15 |

Classification | msc 92B05 |

Classification | msc 92-01 |

Synonym | Poisson distribution |