# independent identically distributed

Two random variables^{} $X$ and $Y$ are said to be *identically distributed* if they are defined on the same probability space^{} $(\mathrm{\Omega},\mathcal{F},P)$, and the distribution function^{} ${F}_{X}$ of $X$ and the distribution function ${F}_{Y}$ of $Y$ are the same: ${F}_{X}={F}_{Y}$. When $X$ and $Y$ are identically distributed, we write $X\stackrel{d}{=}Y$.

A set of random variables ${X}_{i}$, $i$ in some index set^{} $I$, is identically distributed if ${X}_{i}\stackrel{d}{=}{X}_{j}$ for every pair $i,j\in I$.

A collection^{} of random variables ${X}_{i}$ ($i\in I$) is said to be *independent identically distributed*, if the ${X}_{i}$’s are identically distributed, and mutually independent^{} (http://planetmath.org/Independent) (every finite subfamily of ${X}_{i}$ is independent). This is often abbreviated as *iid*.

For example, the interarrival times ${T}_{i}$ of a Poisson process of rate $\lambda $ are independent and each have an exponential distribution^{} with mean $1/\lambda $, so the ${T}_{i}$ are independent identically distributed random variables.

Many other examples are found in statistics^{}, where individual data points are often assumed to realizations of iid random variables.

Title | independent identically distributed |
---|---|

Canonical name | IndependentIdenticallyDistributed |

Date of creation | 2013-03-22 14:27:29 |

Last modified on | 2013-03-22 14:27:29 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 8 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 60-00 |

Synonym | iid |

Synonym | independent and identically distributed |

Defines | identically distributed |