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!
Best Answer
If $Z_1$ and $Z_2$ are random variables with covariance matrix $$\Omega = \left[\begin{matrix}\exp(x)&\exp((x+y)/2)\\\exp((x+y)/2) & \exp(y)\end{matrix}\right]\tag{1}$$ then $Z_1$ and $Z_2$ are perfectly posiitively correlated random variables: their (Pearson) correlation coefficient is $1$ and $$\frac{Z_2-\mu_2}{\exp(y/2)} = \frac{Z_1-\mu_1}{\exp(x/2)}.\tag{2}$$ This holds regardless of the specific distributions of $Z_1$ and $Z_2$ as long as the said distributions have the given means and variances. In particular, it is not necessary to assume that $Z_1$ and $Z_2$ are normal random variables, but doing so is fine.
If $(x,y)$ is a realization of a pair $(X,Y)$ of random variables, then $\Omega$ in $(1)$ can be thought of as the conditional covariance matrix of $Z_1$ and $Z_2$ given that $(X,Y)$ has value $(x,y)$. Thus, the expected value of $\Omega$ regarded as a function of $(X,Y)$ is $$E[\Omega(X,Y)] = \left[\begin{matrix}E[\exp(X)]&E[\exp((X+Y)/2)]\\E[\exp((X+Y)/2)] & E[\exp(Y)]\end{matrix}\right].\tag{3}$$ The diagonal terms of the right side of $(3)$ are just the moment-generating functions $M_X(t)$ and $M_Y(t)$ of $X$ and $Y$ respectively (evaluated at $1$) while the off-diagonal terms are also easily determined in the special case when $(X,Y)$ enjoy a bivariate normal distribution. Note that $X+Y = W$ is a normal random variable with mean $$E[W] = E[X]+E[Y], \quad \operatorname{var}(W) = \operatorname{var}(X) + \operatorname{var}(Y) + 2\operatorname{cov}(X,Y)\tag{4}$$ and so the off-diagonal terms are $M_W(\frac 12)$ where $W$ is a normal random variable with mean and variance given by $(4)$.
Note that the distributions of $Z_1$ and $Z_2$ are irrelevant to these calculations as long as their covariance matrix is given by $(1)$.