independent


In a probability spaceMathworldPlanetmath, we say that the random events A1,,An are independent if

P(Ai1Ai2Aik)=P(Ai1)P(Aik)

for all i1,,ik such that 1i1<i2<<ikn.

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

The random variablesMathworldPlanetmath X1,,Xn are independent if, given any Borel sets B1,,Bn, the random events [X1B1],,[XnBn] are independent. This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to saying that

FX1,,Xn=FX1FXn

where FX1,,FXn are the distribution functionsMathworldPlanetmath of X1,,Xn, respectively, and FX1,,Xn is the joint distribution functionMathworldPlanetmath. When the density functions fX1,,fXn and fX1,,Xn exist, an equivalent condition for independence is that

fX1,,Xn=fX1fXn.

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