# product distribution of two uniform distribution, what about 3 or more

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?
$Z_n = X_1 \cdot X_2 \cdot \ldots \cdot X_n$?

We can at least work out the distribution of two IID ${\rm Uniform}(0,1)$ variables $X_1, X_2$: Let $Z_2 = X_1 X_2$. Then the CDF is Thus the density of $Z_2$ is For a third variable, we would write Then taking the derivative gives In general, we can conjecture that which we can prove via induction on $n$. I leave this as an exercise.