Suppose $X$ and $Y$ follow $N(0,1)$ and $ \operatorname{Corr}(X,Y)=\rho$. Find $ \operatorname{Cov}(X^2,Y^2)$.

Here is what I know $$ \operatorname{Cov}(X^2,Y^2)=E[X^2Y^2]-E[X^2]E[Y^2]$$
Since $E[X^2]= \operatorname{Var}(X)+E^2[X]= \operatorname{Var}(X)$, $\enspace E[X^2]E[Y^2]= \operatorname{Var}(X) \operatorname{Var}(Y)=1$.
But I don't know how to deal with $E[X^2Y^2]$.
Am I in the right track?

  • $\begingroup$ To say that $X\sim N(0,1)$ and $Y\sim N(0,1)$ and $\operatorname{corr}(X,Y) = \text{a particular number}$ is not enough information to specify the distribution of the pair $(X,Y).$ However, it is enough if you add an additional bit of information: that the pair $(X,Y)$ is JOINTLY normally distributed. There is this simple way of getting $X\sim N(0,1)$ and $Y\sim N(0,1)$ and $\operatorname{corr}(X,Y) = \text{your preferred number}$ without $(X,Y)$ being jointly normal: Let $Y=\begin{cases} \phantom{+}X&\text{if } |X|\le c, \\ -X&\text{if }|X| > c, \end{cases}\quad$ and then$\,\ldots \qquad$ $\endgroup$ – Michael Hardy Sep 22 '18 at 21:21
  • $\begingroup$ $\ldots\,$ choose the value of $c$ to to make the correlation what you want it to be. $\qquad$ $\endgroup$ – Michael Hardy Sep 22 '18 at 21:23
  • $\begingroup$ I don't know whether or not the information given, without the assumption of JOINT normality, is enough to determine the correlation between $X^2$ and $Y^2. \qquad$ $\endgroup$ – Michael Hardy Sep 22 '18 at 21:25

Assuming you mean $(X,Y)$ is jointly normal where $X$ and $Y$ have zero means and unit variances and $\operatorname{Corr}(X,Y)=\rho$, we know the conditional distribution of $Y\mid X$, namely

$$Y\mid X\sim N(\rho X,1-\rho^2)$$

Then using the law of total expectation,

\begin{align} E(X^2Y^2)&=E\left[E(X^2Y^2\mid X)\right] \\&=E\left[X^2E(Y^2\mid X)\right] \\&=E\left[X^2\left(\operatorname{Var}(Y\mid X)+(E(Y\mid X))^2\right)\right] \\&=\quad\cdots \end{align}


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