# independent

In a probability space^{}, we say that the random events ${A}_{1},\mathrm{\dots},{A}_{n}$ are
*independent* if

$$P({A}_{{i}_{1}}\cap {A}_{{i}_{2}}\cap \mathrm{\dots}\cap {A}_{{i}_{k}})=P({A}_{{i}_{1}})\mathrm{\dots}P({A}_{{i}_{k}})$$ |

for all ${i}_{1},\mathrm{\dots},{i}_{k}$ such that $$.

An arbitrary family of random events is independent if every finite subfamily is independent.

The random variables^{} ${X}_{1},\mathrm{\dots},{X}_{n}$ are independent if, given any Borel sets ${B}_{1},\mathrm{\dots},{B}_{n}$, the random events $[{X}_{1}\in {B}_{1}],\mathrm{\dots},[{X}_{n}\in {B}_{n}]$ are independent. This is equivalent^{} to saying that

$${F}_{{X}_{1},\mathrm{\dots},{X}_{n}}={F}_{{X}_{1}}\mathrm{\dots}{F}_{{X}_{n}}$$ |

where ${F}_{{X}_{1}},\mathrm{\dots},{F}_{{X}_{n}}$ are the distribution functions^{} of ${X}_{1},\mathrm{\dots},{X}_{n}$, respectively, and ${F}_{{X}_{1},\mathrm{\dots},{X}_{n}}$ is the joint distribution function^{}. When the density functions ${f}_{{X}_{1}},\mathrm{\dots},{f}_{{X}_{n}}$ and ${f}_{{X}_{1},\mathrm{\dots},{X}_{n}}$ exist, an equivalent condition for independence is that

$${f}_{{X}_{1},\mathrm{\dots},{X}_{n}}={f}_{{X}_{1}}\mathrm{\dots}{f}_{{X}_{n}}.$$ |

An arbitrary family of random variables is independent if every finite subfamily is independent.

Title | independent |
---|---|

Canonical name | Independent |

Date of creation | 2013-03-22 12:02:15 |

Last modified on | 2013-03-22 12:02:15 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 11 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 60A05 |