# 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.

• Isn't this just the "product" where the value within the unit cube at x,y,z is just X1(x)*X2(y)*X3(z) where those "X" functions are densities? Or in R notation: dbeta(x, 1,1.5)*dbeta(y,1.5,1)*dbeta(z,2,0.5) – DWin Aug 12 '15 at 1:56
• AKA Dirichlet distribution? – DWin Aug 12 '15 at 2:04
• They are independent, but not identical, I am wondering if they still can be multiplied together. – Deep North Aug 12 '15 at 7:00
• And pdf is not probability, the dbeta(x,1,1.5) is not probability, since Beta distribution is continuous, isn't it? – Deep North Aug 12 '15 at 7:08
• @DeepNorth: Agree densities are not probabilities, but wouldn't the densities of a multivariate distribution where these were uncorrelated be the product of the ednsities times some normalization constant? – DWin Aug 14 '15 at 1:58

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