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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.

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

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  • $\begingroup$ Where did you get the -tlogt +t after the indicator functions? $\endgroup$ – gigglegirl6 Mar 13 at 10:36
  • $\begingroup$ @gigglegirl6 $\int_t^{1} \frac t y \, dy=t\log\, y|_t^{1}=-t\log\, t$ $\endgroup$ – Kavi Rama Murthy Mar 13 at 11:37

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