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independent

In a probability space, we say that the random events A1,…,AnsubscriptA1normal-…subscriptAnA_{1},\dots,A_{n} are independent if

P⁢(Ai1∩Ai2∩…∩Aik)=P⁢(Ai1)⁢…⁢P⁢(Aik)PsubscriptAsubscripti1subscriptAsubscripti2normal-…subscriptAsubscriptikPsubscriptAsubscripti1normal-…PsubscriptAsubscriptikP(A_{{i_{1}}}\cap A_{{i_{2}}}\cap\dots\cap A_{{i_{k}}})=P(A_{{i_{1}}})\dots P(% A_{{i_{k}}})

for all i1,…,iksubscripti1normal-…subscriptiki_{1},\dots,i_{k} such that 1≤i1<i2<⋯<ik≤n1subscripti1subscripti2normal-⋯subscriptikn1\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 X1,…,XnsubscriptX1normal-…subscriptXnX_{1},\dots,X_{n} are independent if, given any Borel sets B1,…,BnsubscriptB1normal-…subscriptBnB_{1},\dots,B_{n}, the random events [X1∈B1],…,[Xn∈Bn]fragmentsfragmentsnormal-[subscriptX1subscriptB1normal-]normal-,normal-…normal-,fragmentsnormal-[subscriptXnsubscriptBnnormal-][X_{1}\in B_{1}],\dots,[X_{n}\in B_{n}] are independent. This is equivalent to saying that

FX1,…,Xn=FX1⁢…⁢FXnsubscriptFsubscriptX1normal-…subscriptXnsubscriptFsubscriptX1normal-…subscriptFsubscriptXnF_{{X_{1},\dots,X_{n}}}=F_{{X_{1}}}\dots F_{{X_{n}}}

where FX1,…,FXnsubscriptFsubscriptX1normal-…subscriptFsubscriptXnF_{{X_{1}}},\dots,F_{{X_{n}}} are the distribution functions of X1,…,XnsubscriptX1normal-…subscriptXnX_{1},\dots,X_{n}, respectively, and FX1,…,XnsubscriptFsubscriptX1normal-…subscriptXnF_{{X_{1},\dots,X_{n}}} is the joint distribution function. When the density functions fX1,…,fXnsubscriptfsubscriptX1normal-…subscriptfsubscriptXnf_{{X_{1}}},\dots,f_{{X_{n}}} and fX1,…,XnsubscriptfsubscriptX1normal-…subscriptXnf_{{X_{1},\dots,X_{n}}} exist, an equivalent condition for independence is that

fX1,…,Xn=fX1⁢…⁢fXn.subscriptfsubscriptX1normal-…subscriptXnsubscriptfsubscriptX1normal-…subscriptfsubscriptXnf_{{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.

Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

60A05 no label found

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Info

Owner: Koro
Added: 2001-12-04 - 04:57
Author(s): Koro

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(v11) by Koro 2013-03-22
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