Condition for Independence from Product of Expectations/Corollary

Corollary to Condition for Independence from Product of Expectations

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

Let $X$ and $Y$ be independent discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.

Then:

$\expect {X Y} = \expect X \expect Y$

assuming the latter expectations exist.

General Result

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

Then:

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

assuming the latter expectations exist.

Proof

From Condition for Independence from Product of Expectations, setting both $g$ and $h$ to the identity functions:

$\forall x \in \R: \map g x = x$

$\forall y \in \R: \map h y = y$

It follows directly that if $X$ and $Y$ are independent, then:

$\expect {X Y} = \expect X \expect Y$

assuming the latter expectations exist.

$\blacksquare$

Note on Converse

Note that the converse of the corollary does not necessarily hold.

Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$ such that:

$\expect {X Y} = \expect X \expect Y$

Then it is not necessarily the case that $X$ and $Y$ are independent.

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 3.3$: Independence of discrete random variables: Theorem $3 \text{C}$