The expected value of the product of two random variables

Summary

• $X$ and $Y$ are two independent random variables . Then

$$E(XY)=E(X)E(Y)$$

• (this is not necessarily true if $X$ and $Y$ are dependent)
• If $f:R→R$ and $g:R→R$ are two arbitrary functions then

$$E\left[ f\left( X \right)g\left( Y \right) \right]=E\left[ f\left( X \right) \right]E\left[ g\left( Y \right) \right]$$

Example

• $X$ and $Y$ are two independent random variables , $E(X^2)=1$ and $E(Y^2)=1$ .
• Then $E(X^2Y^2)=E(X^2)E(Y^2)=1$ .