Covariance of X with Y squared

HardRandom Variables Statistics~8m

Let $(X, Y)$ be bivariate normal with $X$ and $Y$ each standard normal and $\operatorname{Corr}(X, Y) = \rho$ . Compute $\operatorname{Cov}(X, Y^2)$ .

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$\operatorname{Cov}(X,Y^2)=\mathbb{E}[XY^2]$ (since $\mathbb{E}[X]=0$ ).
Write $X=\rho Y+\sqrt{1-\rho^2}Z$ ; the terms are $\rho\,\mathbb{E}[Y^3]$ and $\mathbb{E}[Z]\mathbb{E}[Y^2]$ , both $0$ .

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Covariance of X with Y squared

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