independent

In a probability space, we say that the random events $A_1,\mathrm{\dots },A_n$ areindependent if

$$P(A_{i_1}\cap A_{i_2}\cap \mathrm{\dots }\cap A_{i_k})=P(A_{i_1})\mathrm{\dots }P(A_{i_k})$$

for all $i_1,\mathrm{\dots },i_k$ such that .

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

The random variables $X_1,\mathrm{\dots },X_n$ are independent if, given any Borel sets $B_1,\mathrm{\dots },B_n$, the random events $[X_1\in B_1],\mathrm{\dots },[X_n\in B_n]$ are independent. This is equivalent to saying that

$$F_{X_1,\mathrm{\dots },X_n}=F_{X_1}\mathrm{\dots }{F}_{X_n}$$

where $F_{X_1},\mathrm{\dots },F_{X_n}$ are the distribution functions of $X_1,\mathrm{\dots },X_n$, respectively, and $F_{X_1,\mathrm{\dots },X_n}$ is the joint distribution function. When the density functions $f_{X_1},\mathrm{\dots },f_{X_n}$ and $f_{X_1,\mathrm{\dots },X_n}$ exist, an equivalent condition for independence is that

$$f_{X_1,\mathrm{\dots },X_n}=f_{X_1}\mathrm{\dots }{f}_{X_n}.$$

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