# Condition for Independence from Product of Expectations/Corollary/General Result

## Corollary to Condition for Independence from Product of Expectations

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X_1, X_2, \ldots, X_n$ be independent discrete random variables.

Then:

$\displaystyle \expect {\prod_{k \mathop = 1}^n {X_k} } = \prod_{k \mathop = 1}^n \expect {X_k}$

assuming the latter expectations exist.

## Proof

An inductive proof can be used.