PDF of the product of two independent uniformly distributed random variables

Question

Suppose that X and Y are independent U[0,1]-random variables. Find the probability density function of the product V = XY.

I have seen that ()=(−1)^(−1)log(−1)()/(−1)! for the product of n independent random variables from 0 < z < 1 but I am not sure how to derive this.

Answer 1

$P(XY\leq t)=EP(XY \leq t |Y)=E(\frac t Y I_{Y >t}+I_{Y \leq t})$ so $P(XY\leq t)=t\log (\frac 1 t)+t$ for $0<t<1$. The density is $\log (\frac 1 t)$ for $0<t<1$.