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 \(E(X^2Y^2)=E(X^2)E(Y^2)=1\) .