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.

Tag	`05AW1G0WE1`
URI	`m2://05AW1G0WE1`
Aliases	`multiplicativity-of-expectation`
Kind	statement, theorem
Writing	W0 stub
MSC	60A05
Status	This copy of the site has no build data. Run `m2 status 05AW1G0WE1`.
Uses	Expected value, Independence of two random variables, Independence means the joint law is the product of the marginals

Tag

05AW1G0WE1

URI

m2://05AW1G0WE1

Aliases

multiplicativity-of-expectation

Kind

statement, theorem

Writing

W0 stub

MSC

60A05

Status

This copy of the site has no build data. Run m2 status 05AW1G0WE1.

Uses

Expected value, Independence of two random variables, Independence means the joint law is the product of the marginals

Expectation of a product of independent random variables

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