Let $X_i\sim U(0, 1)$, where $i = 1,\ldots,n$ and where all random variables are independent. Determine using the moment generating function (mgf) the distribution of the random variable $Y$
$$
Y = -2\ln\left(\prod_{i=1}^{n}X_i\right).
$$
I know that in the case variables are independent and they're added together as to form a new variable. The new variable's moment generating function is just the product of the two old variables' mgfs but what confuse me here is that we're inside of the logarithm and I don't know how to deal with that.
Best Answer
\begin{align} M_Y(t) & = \operatorname E(e^{tY}) \\[8pt] & = \operatorname E\left( e^{-2t\ln\prod_{i=1}^n X_i} \right) \\[8pt] & = \operatorname E\left(\left( e^{\ln\prod_{i=1}^n X_i} \right)^{-2t} \right) \\[8pt] & = \operatorname E\left( \left( \prod_{i=1}^n X_i \right)^{-2t} \right) \\[8pt] & = \operatorname E \left( \prod_{i=1}^n X_i^{-2t} \right) \\[8pt] & = \prod_{i=1}^n \operatorname E(X_i^{-2t}) \text{ by independence} \\[8pt] & = \left( \operatorname E\left(X_1^{-2t}\right) \right)^n \text{ by identical distribution}. \end{align}