Distribution over the product of three, or $n$, independent Beta random variables

Question

This is a re-post of a question on the Mathematica stack exchange, as per the advice of another user (see here). I am pursuing a computational solution there, but thought it might be worth looking for a pen-and-paper method.

I would like to calculate the PDF for the product of three independent Beta random variables. Specifically, I would like to find the distribution of the product of the following: $X_1\sim \textrm{Beta}(1,3/2)$, $X_2\sim \textrm{Beta}(3/2,1)$ and $X_3\sim \textrm{Beta}(2,1/2)$.

Does anyone have any idea how I can do this? The reason I state $n$ in the question is because I would like eventually to generalise this calculation to the product of more $\textrm{Beta}$ random variables.

Answer 1

I read this as about the distribution of the product of independent beta distributed random variables. A good tool to study the (distribution of) product of independent random variables is the Mellin transform. The Mellin transform of a beta random variable can be calculated as (an elementary integral) $$ \DeclareMathOperator{\E}{\mathbb{E}} M_X(s) = \E X^s = \frac{B(\alpha+s,\beta)}{B(\alpha,\beta)} $$ where $B( , )$ is the Beta function. The Mellin transform of the (independent) product $X_1 \dotsm X_n$ is the the product of the Mellin transforms $M_{X_i}(s)$.

If that cannot be inverted exactly, the saddlepoint approximation for instance can be used to find a very good approximation for the density. (I will post here some examples, later). This paper finds some general expressions using the Mellin transform approach above, but general solutions do not have a simple form. So the approximation route above could be more practical.