Can somone help me find the PDF of this probability problem

Can somone help me find the PDF of this probability problem. Much help would be appreciated.

(5) Suppose X1, X2, and X3 are independent and uniformly distributed on the interval [0, 1]. Find the PDF for the median of these three random variables.

Solution

Let f(x1, x2, x3) = x1 x2 x3 with x1, x2, x3 ? [0, 1]. For f(x1, x2, x3) to be in the interval [z, z+?) we must have x1 ? z, x2 ? z/x1, and x3 ? [z / (x1 x2), (z+?) / (x1 x2)).

So the probability of Z being between z and z + ? is
?(z to 1) ?(z/x1 to 1) ?(z/(x1 x2) to (z + ?)/(x1 x2) (1) dx3 dx2 dx1
= ?(z to 1) ?(z/x1 to 1) ?/(x1 x2) dx2 dx1
= ?(z to 1) (?/x1) [ln x2][z/x1 to 1] dx1
= ?(z to 1) (?/x1) (-ln (z/x1)) dx1: let u = ln (z/x1), then du = (x1/z) (-z/x1^2) dx1 = -(1/x1) dx1
= ?(0 to ln z) ? u du
= ? (1/2) (ln z)^2
So the pdf of Z is
f(z) = (1/2) (ln z)^2 for z in [0, 1], 0 elsewhere.