Distribution of Product of Exponentiated Log Normal Random Variables

Question

Let the random variable $Y_i$ follow $LOGN(\mu_i, \sigma_i^2)$

Each $Y_i$ is independent.

What is the distribution of the following?

$\prod_{i=1}^{n}Y_i^a$

In other words, what is the distribution of the product of lognormal random variables (each of which with their own, possibly distinct parameters), each of which is raised some arbitrary power?

I'm at a total loss for this. Would somebody be able to provide some direction? Thank you.

Answer 1

Notice that $Y_i = e^{X_i}$ where $X_i$ is a normal random variable. Consequently, $$\prod_{i=1}^{n} Y_i^a = e^{\sum_{i=1}^{n} a X_i}.$$ If $X_i$ are independent then $\sum_{i=1}^{n} a X_i$ is also a Gaussian random variable. Therefore, $\prod_{i=1}^{n} Y_i^a$ is a log-normal random variable.

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Note that the mean and variance of of $\sum_{i=1}^{n} a X_i$ are $a\sum_{i=1}^{n} \mu_{i}$ and $a^2\sum_{i=1}^{n}\sigma_i^2$, respectively.