Probability distribution of the product of random numbers

Question

For applied mathematics to evolutionary biology I am often faced to have to describe a probability distribution function (PDF) which results from the product of a function in which a parameter is drawn from a PDF. For example the random variable for which I'd like to describe the PDF is $Y$ such as

$$Y = \prod_{i=1}^{n} f(x_i)$$

, where each $x_i$ is drawn from a known PDF. Do you have some kind of general hints/advice for solving this kind of issue? If general advice are not possible, below I am suggesting two simple (or at least I hope they are simple) examples of problems:

• Find the PDF of $Y$ such as $$Y = \prod_{i=1}^n x_i$$, where each $x_i$ is a value drawn from an exponential distribution with parameter $\lambda$. Below is the exponential distribution:

$$Pr(X=x) = \lambda e^{-\lambda x}$$

• Find the PDF of $Y$ such as $$Y = \prod_{i=1}^n log_e(x_i)^2$$, where each $x_i$ is a value drawn from an gaussian distribution with mean $\mu$ and variance $\sigma ^2$. Below is the gaussian distribution:

$$Pr(X=x) = \frac{1}{\sigma \sqrt {2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma ^2}}$$

Answer 1

As a general advice, you are looking for products of random variables (or, more generally, products of functions of random variables). To determine the distribution (pdf, cdf) of a product of random variables, different techniques may apply.

You could look here: What is the distribution of a random variable that is the product of the two normal random variables ? (look at all of the answers)

Or here: http://www.maths.bris.ac.uk/~macpd/georgiou/ProductRVs%20revised.pdf

Answer 2

What about taking the logarithm of the product? Then you have a sum of random variables. Depending on the specifics, the central limit theorem (CLT) could apply to $\log Y$, that is, $\log Y$ could be normal, which would imply that $Y$ has a log-normal distribution.

Since here all your $X_i$'s have an exponential distribution with the same parameter $\lambda$ and are independent, the CLT does actually apply, after normalizing the variables. Let $Z_n = \log Y_n$ with $Y_n$ being the product over the first $n$ factors. Also, let

$$Z_n^* = \frac{Z_n - \mbox{E}(Z_n)}{\sqrt{\mbox{Var}(Z_n)}}.$$

Then $Z_n^*$ is normal and $\exp Z_n^*$ is log-normal as $n\rightarrow \infty$.