PDF of $-\sum\limits_{i=1}^n\log(X_i)$ for $(X_i)$ i.i.d. Beta$(\theta,1)$

Question

Suppose that $X_1, ..., X_n$ are random variables independent and identically distributed with beta$(\theta,1)$ distribution and that $\log$ is the natural logarithm, then what is the PDF of $$Y=-\sum\limits_{i=1}^n\log(X_i)\ ?$$

I have no idea of how to solve it. I tried use the caracteristic function(Moment generating function), but I did not get any results.

$\phi_Y (t) = E( e^{t(-\sum_{i=1}^{n} \log (X_i))} ) = E( e^{(\sum_{i=1}^{n} \log (X_i^{-t}))} ) =\displaystyle E (\prod_{i=1}^{n} X_i^{-t})$

I was hoping to get into some known generated function.

Answer 1

The transformed random variables $X_i^\theta$ are i.i.d. uniformly distributed on $[0,1]$, so $\theta Y$ is distributed as the sum of $n$ standard exponential random variables, that is, Gamma distributed with parameter $n$.