Covariance between squared and exponential of Gaussian random variables

Question

Assuming the random vector $[X \ \ Y]'$ follows a bivariate Gaussian distribution with mean $[\mu_X \ \ \mu_Y]'$, and covariance matrix $ \left[ \begin{array}{cc} \sigma_X ^2 & \sigma_Y \ \sigma_X \ \rho \\ \sigma_Y \ \sigma_X \ \rho & \sigma_Y^2 \end{array} \right]$, I am looking for the expression of $\text{Cov}(X^2, \exp{Y})$.

I know there exists a formula (Stein's lemma) for $\text{Cov}(X, \exp{Y})$ and $\text{Cov}(X^2, {Y})$, but I did not manage to find a closed form formula for $\text{Cov}(X^2, \exp{Y})$ given the two transformations.

Answer 1

Under the given assumptions, $(X,Y)$ have a bivariate Normal distribution with joint pdf $f(x,y)$:

You seek:

Notes

1. The Cov function used above to automate the calculation of the covariance is from the mathStatica package for Mathematica. As disclosure, I should add that I am one of the authors.