Expectation of a product of independent random variables

Statement

Let $X$ and $Y$ be independent integrable real random variables. Then $XY$ is integrable and $$E[XY] = E[X] \, E[Y].$$

Remarks

The proof uses that the joint law of $(X, Y)$ is the product of the marginal laws, and then Fubini's theorem. For nonnegative independent $X$ and $Y$, the equality holds in $[0, \infty]$ without integrability. The converse is false: $E[XY] = E[X] \, E[Y]$ means only that $X$ and $Y$ are uncorrelated. More generally, $E[g(X) \, h(Y)] = E[g(X)] \, E[h(Y)]$ for measurable $g$ and $h$ with $g(X)$ and $h(Y)$ integrable.

Explanations

L1 human

For independent variables, the average of a product is the product of the averages. For two fair dice, the expected product of the two results is $3.5 \cdot 3.5 = 12.25$.