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

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.

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 $$M_{X_i}(s)$$.