Find $\operatorname{corr}(X^2,Y^2)$ using conditional expectation

Question

I'm stuck and I think I need some help.

Suppose that $(X,Y )$ is bivariate normal. Both $X$ and $Y$ have mean $0$ and variance $1$. Assume $\operatorname{corr}(X,Y) = \rho$. Find $\operatorname{corr}(X^2,Y^2)$. Hint: Use conditional expectations.

Here's what I've tried so far, but I think I'm not even close:

Thanks!

Answer 1

First, as suggested, by conditioning you get

$$\mathbb{E}[X^2Y^2]=\mathbb{E}\{\mathbb{E}[X^2Y^2|X]\}=\mathbb{E}\{X^2\mathbb{E}[Y^2|X]\}=$$

$$\mathbb{E}\{X^2[\mathbb{V}(Y|X)+\mathbb{E}^2(Y|X)]\}=\mathbb{E}[(1-\rho^2)X^2+\rho^2X^4]$$

Remember that, for a Standard Gaussian, $\mathbb{E}[X^{2n}]=\frac{(2n)!}{2^n\cdot n!}\rightarrow \mathbb{E}[X^4]=3$,

you have

$$\mathbb{E}[X^2Y^2]=1+2\rho^2$$

Now remember that $X^2\sim \chi_{(1)}^2$ thus

$$Corr(X^2,Y^2)=\frac{2\rho^2}{\sqrt{2\times2}}=\rho^2$$