# Thread: Correlation of bivariate normal

1. ## Correlation of bivariate normal

Hi, I am having a lot of trouble with this problem and would appreciate any type of help!

So X and Y are bivariate normal with both means = 0, variances = 4, and correlation p. I need to find $\displaystyle corr(X^2, Y^2)$

To find correlation, I would start off finding covariance. I know that $\displaystyle Cov(X^2, Y^2) = E(X^2 Y^2)-E(X^2) E(Y^2)$
I know that $\displaystyle E(X^2)=E(Y^2)=4$ but I don't know how to figure out $\displaystyle E(X^2 Y^2)$.

Or maybe I'm completely approaching this problem wrong?

Thanks!

2. Hello,

I can't see any other way than writing down the pdf of the bivariate normal (see wikipedia if you don't know it), then use the Jacobian transformation to get the joint distribution of $\displaystyle (X^2Y^2,Y)$ for example, and then integrate with respect to the second variable.

3. Thanks, I am trying to approach it as you suggested. But I get a messy integral that doesn't simplify into a marginal. For instance, I set U = XY and V = Y. I find their joint distribution, but I can't integrate with respect to V to find the marginal distribution of U.

4. Okay I agree, it's not nice at all

I'm sorry, I just saw this in the wikipedia : http://en.wikipedia.org/wiki/Multiva...Higher_moments

which refers to http://en.wikipedia.org/wiki/Isserlis%E2%80%99_theorem : consider the vector $\displaystyle (X,X,Y,Y)$ (it's still a multivariate normal distribution). By the theorem, you get $\displaystyle E[X^2Y^2]=E[X X]E[YY]+E[XY]E[XY]+E[XY]E[XY]=4\times 4+2[\text{cov}(X,Y)]^2$ =...
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