# 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}.

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.

Title independent Independent 2013-03-22 12:02:15 2013-03-22 12:02:15 Koro (127) Koro (127) 11 Koro (127) Definition msc 60A05