Find $E[X^2Y^2]$ if you know $E[XY]$, correlation $\rho_{XY}$ and distribution of $X,Y$

Question

I have two random variables, both have Bernoulli distribution with mean $p = 0.5$. Next I know that their correlation is $\rho_{XY} = -0.02$. From that I found $E[XY] = 0.2$. Next I have to find $E[X^2Y^2]$, but I have no idea how.

I don't have joint probability of the variables, which would make it an easy problem. I had an idea to use identity $E[X^2Y^2] = D[XY] + (E[XY])^2$, but I dont know how to find D[XY], without said joint probability. What am I missing?

Answer 1

If $X$ and $Y$ take only the values $0$ and $1$ then $X^{2}=X$ and $Y^{2}=Y$. (Because $0^{2}=0$ and $1^{2}=1$). Hence $E[X^{2}Y^{2}]=E[XY]=0.2$.