Suppose I have a sequence of iid random variables $X_1, \ldots, X_n$ following the pdf:
$$ f_\theta (x) = \theta x^{\theta-1} $$
for $\theta >0$ and $0 <x<1$.
I would like to find the distribution of:
$$ \prod_{i=1}^n X_i $$
Is there an easy way to do this? I know one method is to first find the distribution of $X_1X_2$, then find the product distribution of $X_1X_2$ and $X_3$. Is there an easier way than doing that?
A simple approach is to consider the PDF of the transformed random variable $$Y_i = -\log X_i,$$ and conclude $Y_i \sim \operatorname{Exponential}(\theta)$, where $\theta$ is a rate parameter. Consequently, $$\prod_{i=1}^n X_i = \exp\left(-\sum_{i=1}^n Y_i\right),$$ and we can now exploit the fact that the sum of iid exponential random variables is gamma distributed; i.e., $$\sum_{i=1}^n Y_i \sim \operatorname{Gamma}(n,\theta).$$ The final step is to transform the gamma PDF to recover the product distribution. I have left the details as an exercise.
