# What is the covariance of the exponential of two jointly distributed normal random variables?

Suppose I have the bivariate normal variable $$[X \ \ Y]'$$ which has mean $$[\mu_X \ \ \mu_Y]'$$ and covariance matrix $$\left[ \begin{array}{cc} \sigma_X ^2 & \sigma_Y \ \sigma_X \ \\ \sigma_Y \ \sigma_X \ & \sigma_Y^2 \end{array} \right]$$.

I am trying to figure out what is $$\text{Cov}(e^X, e^Y)$$.

I'm aware of Stein's lemma which says $$\text{Cov}(g(X), Y) = \text{Cov}(X,Y)E[g'(X)]$$, but I don't know how it's derived, or how to extend it to functions on both $$X$$ and $$Y$$. Presumably this case should be simpler since it's the exponential function.

Otherwise I'm unsure what other steps to take. I don't need to prove this, but the answer is necessary for a calculation I am doing.

• Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Commented Mar 5, 2021 at 3:23
• Thanks, I appreciate the tips! I've updated the question with more information, I hope this improves it. Commented Mar 5, 2021 at 3:30

$$\operatorname{Cov}[e^X, e^Y] = \operatorname{E}[e^{X}e^{Y}] - \operatorname{E}[e^X]\operatorname{E}[e^Y] = \operatorname{E}[e^{X+Y}] - M_X(1) M_Y(1),$$ where $$M_X, M_Y$$ are the moment-generating functions of the marginal distributions of $$X$$ and $$Y$$. The handling of $$\operatorname{E}[e^{X+Y}]$$ requires an additional step, which is to compute the mean and variance of $$X+Y$$, since their sum is normal whenever $$(X,Y)$$ are bivariate normal. Specifically, we obviously have $$\mu = \operatorname{E}[X+Y] = \mu_X + \mu_Y$$ and $$\sigma^2 = \operatorname{Var}[X+Y] = \sigma_X^2 + 2\sigma_{XY} + \sigma_Y^2.$$ This results in a third MGF to compute. These are all straightforward calculations if one is familiar with bivariate normal distributions.