product distribution of two uniform distribution, what about 3 or more

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=X1X2?

What if there are 3; Z3=X1X2X3?

What if there are n of such uniform variables?
Zn=X1X2Xn?

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[Z2z]=1x=0Pr[X2z/x]fX1(x)dx=zx=0dx+1x=zzxdx=zzlogz. Thus the density of Z2 is fZ2(z)=logz,0<z1. For a third variable, we would write FZ3(z)=Pr[Z3z]=1x=0Pr[X3z/x]fZ2(x)dx=zx=0logxdx1x=zzxlogxdx. Then taking the derivative gives fZ3(z)=12(logz)2,0<z1. In general, we can conjecture that fZn(z)={(logz)n1(n1)!,0<z10,otherwise, which we can prove via induction on n. I leave this as an exercise.

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

Leave a Comment