Product of independent uniform random variables ($f_{XY}(z)=b^{-2}\log (b^2/z),0<z<b^2$)

For independent random variables $X,Y$ with the uniform distribution on $(0,b)$, use the probability density of a product of random variables and integrate $1/(b^{2}x)$ over $z/b<x<b$. The displayed probability density function is zero off $(0,b^2)$ and has integral one. More generally, for independent and identically distributed factors $U_j$ with this uniform distribution, $W_n={\prod }_{j=1}^n{U}_j$ satisfiesThe expectation of a product of independent random variables proves these identities directly.