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.,

What if there are 3; Z3=X1X2X3?

What if there are n of such uniform variables?


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.

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

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