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

Question

Is it possible to calculate $\mathbb{E}[X^2 Y]$, where $X$ and $Y$ are standard normal random variables with a correlation $\rho$?

I can't do it using covariance or a law of iterated expectations, since I can`t find $\mathbb{E}[X^2\mid Y]$...

Answer 1

Assuming $X$ and $Y$ are jointly normal, $E[X^2Y]=E[E[X^2Y\mid Y]] = E[YE[X^2\mid Y]]$. But now $E[X^2\mid Y]=\operatorname{Var}[X\mid Y]+E[X\mid Y]^2 = (1-\rho^2) + \rho^2 Y^2$ Thus the answer is $E[(1-\rho ^2)Y+\rho Y^3]=0+0=0$

Note that $E[Y^3]=0$ as $Y^3$ and $-Y^3$ have the same distribution (and are absolutely integrable).

If $X$ and $Y$ are not assumed to be jointly normal, there should be an interval (depending on $\rho$ and including $0$) such that you can make $E[X^2Y]$ be anything in that interval. To see why this can be non-zero you can arrange for $Y$ to be positive when $|X|$ is large and for $Y$ to be negative when $|X|$ is small.