Question

Distribution of the product of two uniform random variables

Answer 1

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]=∫

1

x=0

Pr[X2≤z/x]fX1(x)dx =∫

z

x=0

dx+∫

1

x=z

z

x

dx =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]=∫

1

x=0

Pr[X3≤z/x]fZ2(x)dx =−∫

z

x=0

logxdx−∫

1

x=z

z

x

logxdx.

Then taking the derivative gives

fZ3(z)=

1

2

(logz)2,0<z≤1.

In general, we can conjecture that

fZn(z)={

(−logz)n−1

(n−1)!

, 0<z≤1 0, otherwise,

which we can prove via induction on n. I leave this as an exercise.