independent
In a probability space, we say that the random events A1,…,An are
independent if
P(Ai1∩Ai2∩…∩Aik)=P(Ai1)…P(Aik) |
for all i1,…,ik such that 1≤i1<i2<⋯<ik≤n.
An arbitrary family of random events is independent if every finite subfamily is independent.
The random variables X1,…,Xn are independent if, given any Borel sets B1,…,Bn, the random events [X1∈B1],…,[Xn∈Bn] are independent. This is equivalent
to saying that
FX1,…,Xn=FX1…FXn |
where FX1,…,FXn are the distribution functions of X1,…,Xn, respectively, and FX1,…,Xn is the joint distribution function
. When the density functions fX1,…,fXn and fX1,…,Xn exist, an equivalent condition for independence is that
fX1,…,Xn=fX1…fXn. |
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 |