# PDF of the product of two independent uniformly distributed random variables

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

$$P(XY\leq t)=EP(XY \leq t |Y)=E(\frac t Y I_{Y >t}+I_{Y \leq t})$$ so $$P(XY\leq t)=t\log (\frac 1 t)+t$$ for $$0. The density is $$\log (\frac 1 t)$$ for $$0.
• @gigglegirl6 $\int_t^{1} \frac t y \, dy=t\log\, y|_t^{1}=-t\log\, t$ – Kavi Rama Murthy Mar 13 at 11:37