Could someone help me to solve that exercise?
If X and Y are discrete random variables, each taking only two distinct values, prove that X and Y are independent if and only if $\mathbf{E} $(X) $\mathbf{E} $(Y) = $\mathbf{E} $(XY)
The necessity of $\mathbf{E} $(X) $\mathbf{E} $(Y) = $\mathbf{E} $(XY) is immediate, hence I would like to know how to prove the sufficiency.
Hint: call $P(X=x_1)=p$, $P(Y=y_1)=q$, $P(XY=x_1y_1)=a$. From the given condition it follows by brute force that $(a-pq)(x_1-x_2)(y_1-y_2)=0$.