TITLE: product distribution of two uniform distribution, what about 3 or more
QUESTION [61 upvotes]: Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$.
What is the product distribution of two of such random variables, e.g.,
$Z_2 = X_1 \cdot X_2$?
What if there are 3; $Z_3 = X_1 \cdot X_2 \cdot X_3$?
What if there are $n$ of such uniform variables?
$Z_n = X_1 \cdot X_2 \cdot \ldots \cdot X_n$?

REPLY [12 votes]: An adaptation of this answer is given here.

PDF of a Function of a Random Variable
If $P(X\le x)=F(x)$ is the CDF of $X$ and $P(Y\le y)=G(y)$ is the CDF of $Y$ where $Y=f(X)$, then 
$$
F(x)=P(X\le x)=P(Y\le f(x))=G(f(x))\tag1
$$
Taking the derivative of $(1)$, we get
$$
F'(x)=G'(f(x))\,f'(x)\tag2
$$
where $F'$ is the PDF of $X$ and $G'$ is the PDF of $Y$.

PDF of the Product of Independent Uniform Random Variables
If $[0\le x\le1]$ is the PDF for $X$ and $Y=\log(X)$, then by $(2)$ the PDF of $Y$ is $e^y[y\le0]$. The PDF for the sum of $n$ samples of $Y$ is the $n$-fold convolution of $e^y[y\le0]$ with itself. The Fourier Transform of this $n$-fold convolution is the $n^\text{th}$ power of the Fourier Transform of $e^y[y\le0]$, which is
$$
\int_{-\infty}^0 e^{-2\pi iyt}e^y\,\mathrm{d}y=\frac1{1-2\pi it}\tag3
$$
Thus, the PDF for the sum of $n$ samples of $Y$ is
$$
\begin{align}
\sigma_n(y)
&=\int_{-\infty}^\infty\frac{e^{2\pi iyt}}{(1-2\pi it)^n}\,\mathrm{d}t\tag{4a}\\
&=\frac{e^y}{2\pi i}\int_{1-i\infty}^{1+i\infty}\frac{e^{-yz}}{z^n}\,\mathrm{d}z\tag{4b}\\
&=e^y\frac{(-y)^{n-1}}{(n-1)!}\,[y\le0]\tag{4c}
\end{align}
$$
Explanation:
$\text{(4a)}$: take the inverse Fourier Transform
$\text{(4b)}$: substitute $t=\frac{1-z}{2\pi i}$
$\text{(4c)}$: if $y\gt0$, close the contour on the right half-plane, missing the singularity at $z=0$
$\phantom{\text{(4c):}}$ if $y\le0$, close the contour on the left half-plane, enclosing the singularity at $z=0$  
We can get the PDF for the product of $n$ samples of $X$ by applying $(2)$ to $(4)$
$$
\bbox[5px,border:2px solid #C0A000]{\pi_n(x)=\frac{(-\log(x))^{n-1}}{(n-1)!}\,[0\le x\le1]}\tag5
$$