I am struggling with a proof, and I am wondering if anyone can help or point me to the right direction. Suppose that we have two variables, $X$ and $Y$, and they follow a multivariate normal distribution with covariance matrix $Q$. Now, suppose that we have two more variables, $Z_1$ and $Z_2$ that follow a normal distribution with covariance $\Omega$, whose elements are,
$\Omega_{11} = exp(X)$, $\Omega_{22} = exp(Y)$, $\Omega_{21} = \Omega_{12}=\exp (X/2)\exp(Y/2)$,
And the question is, what is the expectation of $\Omega$?
For $\Omega_{11}$ and $\Omega_{22}$ it is clear that the expectation is $\exp(0.5Q_{11})$ and $\exp(0.5Q_{22})$ using the properties of log normal variables, but I am not sure how to compute it for the covariance term.
Thanks!