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# independent

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

$P(A_{{i_{1}}}\cap A_{{i_{2}}}\cap\dots\cap A_{{i_{k}}})=P(A_{{i_{1}}})\dots P(% A_{{i_{k}}})$ |

for all $i_{1},\dots,i_{k}$ such that $1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n$.

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

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

$F_{{X_{1},\dots,X_{n}}}=F_{{X_{1}}}\dots F_{{X_{n}}}$ |

where $F_{{X_{1}}},\dots,F_{{X_{n}}}$ are the distribution functions of $X_{1},\dots,X_{n}$, respectively, and $F_{{X_{1},\dots,X_{n}}}$ is the joint distribution function. When the density functions $f_{{X_{1}}},\dots,f_{{X_{n}}}$ and $f_{{X_{1},\dots,X_{n}}}$ exist, an equivalent condition for independence is that

$f_{{X_{1},\dots,X_{n}}}=f_{{X_{1}}}\dots f_{{X_{n}}}.$ |

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

## Mathematics Subject Classification

60A05*no label found*

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## Attached Articles

## Corrections

typo by yark ✓

typo by ratboy ✓

capitalization by Mathprof ✓

mutual independence by CWoo ✓