Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas.
Consider the corresponding log-normal random variables: $Z_1 = \exp(X_1)$, $Z_2 = \exp(X_2)$.
Question: what is the distribution of the product of the two random variables, i.e., the distribution of $Z_1Z_2$?
If the normal random variables $X_1, X_2$ are independent, or they have a bivariate normal distribution, the answer is simple: we have $Z_1Z_2 = \exp(X_1+X_2)$ with the sum $X_1+X_2$ normal, hence the product $Z_1Z_2$ is still lognormal.
But suppose that $X_1, X_2$ are generally $not$ independent, say with correlation $\rho$. What can we say about the distribution of $Z_1Z_2$?
Best Answer
Using Dilips answer here, if $X$ and $Y$ are bi-variate normal and $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$ and the correlation between $X$ and $Y$ is $\rho$. Then
$$ Cov(X,Y) = \rho \sigma_1 \sigma_2,$$
$$X + Y \sim N(\mu_1 + \mu_2, \sigma^2_1 + \sigma^2_2 + 2\rho\sigma_1 \sigma_2). $$
Thus $Z_1Z_2$ will also be a lognormal distribution with parameters $\mu_1 + \mu_2$ and $\sigma^2_1 + \sigma^2_2 + 2\rho\sigma_1 \sigma_2$.