Question:
Given a sequence of i.i.d. Uniform(0,1) random variables $(X_1, ... , X_n)$, show that the probability density function of $Y = \prod_{i=1}^{n} X_i$ is given by $f_{Y} (y) = \frac{(-\log{(y)})^{n-1}}{(n-1)!}, y \in (0,1)$.
Attempt:
In the earlier parts of the question, I proved that -log($X_i$) ~ exp(1), and that $X_i + X_j$ ~ Gamma(2, $\lambda$), so I am guessing I need to use these facts, but I'm not sure how to proceed. Thanks for any help.
