Solved – Expected values of (X*Y^2) when X and Y are dependent normally distributed RVs

I have two jointly normal variables X and Y with mean both zeros and variances $sigma^2_{X}$ and $sigma^2_{Y}$ separately, the covariance is $sigma_{XY}$. Now I want to calculate the expected value of $Z=X*Y^{2}$, $E(Z)$. Any ideas?


$E[XY^2] = E[ E[XY^2/ Y] ] = E[Y^2 E[Xmid Y]]=alpha E[Y^3 ] =0$

$alpha = frac{sigma_X}{sigma_Y} rho$

$rho = cor(X,Y)$

The thing is : your expectation is an integral of an odd $0$-symetric function on $[-infty, +infty] $ this is why it's equal to zero

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