6
$\begingroup$

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 $Z_1Z_2$?

$\endgroup$
  • $\begingroup$ This might be useful: stats.stackexchange.com/questions/19948/… $\endgroup$ – Greenparker May 15 '16 at 17:20
  • $\begingroup$ I doubt it though.. basically this question asks "if the marginals are normally distributed, can we say anything about their joint distribution?" And I don't think we can say much in general $\endgroup$ – Ant May 15 '16 at 19:27
  • 1
    $\begingroup$ $Z_1 Z_2 = \exp(X_1 + X_2)$ in general, so your real question is whether $X_1+X_2$ is normal (which it will be if $X_1, X_2$ are bivariate normal with correlation $\rho$) $\endgroup$ – Henry May 15 '16 at 22:12
  • 3
    $\begingroup$ If you don't have bivariate normality, merely specifying the correlation and the margins is not sufficient to pin down the bivariate distribution. $\endgroup$ – Glen_b May 16 '16 at 1:31
3
$\begingroup$

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 $\sigma^2_1 + \sigma^2_2 + 2\rho\sigma_1 \sigma_2$.

$\endgroup$
  • 2
    $\begingroup$ Add some qualification? This is true if the distribution of (X,Y) is bi-variate normal, but that the marginal distribution of X is normal and the marginal distribution of Y is normal does not imply the joint distribution is bi-variate normal... stats.stackexchange.com/questions/30159/… $\endgroup$ – Matthew Gunn May 15 '16 at 17:41
  • 1
    $\begingroup$ @MatthewGunn Interesting. I was blissfully unaware of this. As it stands, my answer does not address the question completely. I will wait a couple of hours before deleting it. Thanks. $\endgroup$ – Greenparker May 15 '16 at 17:44
  • $\begingroup$ Do you have alot of data from this distribution? In such case you can plot it, or otherwise test if bivariate normal is a good distribution approximation. Otherwise, maybe you could estimate a copula and go from there! $\endgroup$ – kjetil b halvorsen May 15 '16 at 18:36
  • $\begingroup$ This is quite obvious, but you're missing the point: what happens if you don't know whether the two marginals have a bivariate distribution? $\endgroup$ – RandomGuy May 16 '16 at 10:53
  • $\begingroup$ Thanks! This amazing answer saved my day! $\endgroup$ – Jinhua Wang May 26 at 15:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.