Expectation of normal and log normal distribution

Question

Let $X \sim N(\mu_x, \sigma_x^2)$ and $Y\sim N(\mu_y, \sigma_y^2)$, with correlation $\rho$.

How do I find $$E[Xe^Y]$$?

I tried a bunch of things without result. I'm also interested in "general" methods if they exists

Thank you!

Answer 1

Presumably $X$ and $Y$ are jointly normal. Then $$E[X|Y] = \mu_x + \dfrac{\rho \sigma_x}{\sigma_y} (Y - \mu_y)$$ and then $$ E[X e^Y] = E[E[X|Y] e^Y] = \ldots $$ Now use the moment generating function for $Y$: $$E[e^{tY}] = \exp(t \mu_y + t^2 \sigma_y^2/2)$$ and its derivative with respect to $t$, which is $E[Y e^{tY}]$.