Can we find $E(XY)$ if given $E(X), E(Y), E(X^2), E(Y^2)$?

Question

Given $X$ and $Y$ are two random variables with $E(X)=a, E(Y)=b, E(X^2)=c, E(Y^2)=d$. $X$ and $Y$ not necessary to independent. I have problem, I can't find E(XY) when given $E(X), E(Y), E(X^2), E(Y^2)$. I try to find using $$E((X+Y)^2)=E(X^2)+2E(XY)+E(Y^2),$$ But I confused to find $E((X+Y)^2)$. So can we find E(XY) if only given $E(X), E(Y), E(X^2), E(Y^2)$?

Answer 1

Let $X$, $Y$ be uniformly distributed random variables, over $[0;1]$, so $\mathsf E(X)=\mathsf E(Y)=\tfrac 12$ and $\mathsf E(X^2)=\mathsf E(Y^2)=\tfrac 13$.

Consider the following cases.

• $X,Y$ are independent. So $\mathsf E(XY)-\mathsf E(X)\mathsf E(Y)=0$, because they are uncorrelated.$$\text{Case 1: }\mathsf E(XY)=\tfrac 14$$

• $X=Y$ almost surely. So $\tfrac{\mathsf E(XY)-\mathsf E(X)\mathsf E(Y)}{\sqrt{(\mathsf E(X^2)-\mathsf E(X)^2)(\mathsf E(Y^2)-\mathsf E(Y)^2)}}=1$.$$\text{Case 2: }\mathsf E(XY)=\tfrac 13$$

• $X=1-Y$ almost surely. So $\tfrac{\mathsf E(XY)-\mathsf E(X)\mathsf E(Y)}{\sqrt{(\mathsf E(X^2)-\mathsf E(X)^2)(\mathsf E(Y^2)-\mathsf E(Y)^2)}}=-1$.$$\text{Case 3: }\mathsf E(XY)=\tfrac 16$$

Thus the four expectations: $\mathsf E(X),\mathsf E(Y),\mathsf E(X^2),$ and $\mathsf E(Y^2)$, are not sufficient to determine what $\mathsf E(XY)$ will be.   You also need to know how they are correlated.

Answer 2

Suppose $X=Y$ and that $X$ is half the time $1$ and half the time $-1$. Then $E(X)=E(Y)=0$ and $E(X^2)=E(Y^2)=1$. Also $E(XY)=1$.

Now replace $Y$ with $-X$. You haven't changed any of $E(X),E(Y),E(X^2)$ or $E(Y^2)$, but you've changed $E(XY)$ from $1$ to $-1$.

So $E(XY)$ is not determined by the other four.