Calculate $E(XY)$ if $\mathrm{Cov}(X, Y) = 0$

Question

$E(X) = 3, E(Y) = 4$ and $E(X^2) = 10, E(Y^2) = 25$

How can I calculate $E(XY)$ If I know that $\mathrm{Cov}(XY) = 0$

I know that If $X$ and $Y$ are independent then $\mathrm{Cov}(X, Y) = 0.$ But zero covariance does not always imply independence

Answer 1

Use the property $\mathrm{Cov}[X,Y] =E[XY]-E[X]E[Y].$ This gives $\mathrm{Cov}[XY]=E[XY]-12\Rightarrow E[XY]=12.$

Proof of property: $\mathrm{Cov[X,Y]}=E[(X-E[X])(Y-E[Y])]\\=E[XY]-E[E[X]Y]-E[XE[Y]]+E[E[X]E[Y]]\\=E[XY]-E[X]E[Y]-E[X]E[Y]+E[X]E[Y]\\=E[XY]-E[X]E[Y].$