Say X1,X2,…,Xn 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.,

Z2=X1⋅X2?What if there are 3; Z3=X1⋅X2⋅X3?

What if there are n of such uniform variables?

Zn=X1⋅X2⋅…⋅Xn?

**Answer**

We can at least work out the distribution of two IID Uniform(0,1) variables X1,X2: Let Z2=X1X2. Then the CDF is FZ2(z)=Pr[Z2≤z]=∫1x=0Pr[X2≤z/x]fX1(x)dx=∫zx=0dx+∫1x=zzxdx=z−zlogz. Thus the density of Z2 is fZ2(z)=−logz,0<z≤1. For a third variable, we would write FZ3(z)=Pr[Z3≤z]=∫1x=0Pr[X3≤z/x]fZ2(x)dx=−∫zx=0logxdx−∫1x=zzxlogxdx. Then taking the derivative gives fZ3(z)=12(logz)2,0<z≤1. In general, we can conjecture that fZn(z)={(−logz)n−1(n−1)!,0<z≤10,otherwise, which we can prove via induction on n. I leave this as an exercise.

**Attribution***Source : Link , Question Author : lulu , Answer Author : heropup*