Prove that \text{Corr}(X^2,Y^2)=\rho^2\text{Corr}(X^2,Y^2)=\rho^2 where X,YX,Y are jointly N(0,1)N(0,1) variables with correlation \rho\rho

Consider jointly normally distributed random variables X,Y\sim N(0,1) that have \text{Corr}(X,Y)=\rho.
Show that \text{Corr}(X^2,Y^2)=\rho^2.

(Hint: Consider X,U\sim N(0,1) where they are independent then we have Y=\rho X+\sqrt{1-\rho^2}U).

I sense that a transform is required for both X and Y but not sure where to go next.

edit: I am more interested in knowing how the hint is used.

Answer

Hint:

Let (X,Y) be jointly normal variables with zero means and unit variances and \text{Corr}(X,Y)=\rho.

By definition, \begin{align}\text{Corr}(X^2,Y^2)=\frac{\text{Cov}(X^2,Y^2)}{\sqrt{\text{Var}(X^2)\text{Var}(Y^2)}}\end{align}

where \text{Cov}(X^2,Y^2)=\mathbb E(X^2Y^2)-\mathbb E(X^2)\mathbb E(Y^2), and

\text{Var}(X^2)=\mathbb E(X^4)-(\mathbb E(X^2))^2=\text{Var}(Y^2).

For finding \mathbb E(X^2Y^2) quickly, note that \mathbb E(X^2Y^2)=\mathbb E(\mathbb E(X^2Y^2\mid X))=\mathbb E(X^2\,\mathbb E(Y^2\mid X)).

And we know that Y\mid X\sim\mathcal{N}(\rho X,1-\rho^2).

So, \mathbb E(Y^2\mid X)=\text{Var}(Y\mid X)+(\mathbb E(Y\mid X))^2=\cdots.

I think you can find the moments now.

Attribution
Source : Link , Question Author : Brofessor , Answer Author : StubbornAtom

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