Statistics 2 question

Question

Let (X, Y ) have the joint pdf fX,Y (x, y) = ( 2(2x + y)/3, 0 < x < 1, 0 < y < 1,

0, otherwise.

Find the pdf of Z = XY

Answer 1

The area such that Z <= z includes the rectangle [0, z] x [0,1] and the region with boundary points [z, 0], [z, 1], [1, z]. and y = z/x for x from 0 to 1.

Using I[a,b] for the integral from a to b and E[a,b] for evaluation from a to b,

I[0,z]I[0,1] 4x/3 + 2y/3 dy dx = I[0,z] 4xy/3 + y²/3 E[0,1] dx =

I[0,z] 4x/3 + 1/3 dx = 2x²/3 + 1/3x E[0,z] = 2z²/3 + 1/3z

I[z,1]I[0,z/x]4x/3+2y/3 dy dx =

I[z,1] 4xy/3 + y²/3 E[0, z/x] dx =

I[z,1] 4z/3 + z²/3x² dx = 4zx/3 - z²/3x E[z,1] =

4z/3 - z²/3 -4z²/3 + z/3

Adding these together, we get 2z²/3 + 1/3z + 4z/3 - z²/3 -4z²/3 + z/3 =

2z-z²

^{As a check, note that this equals 0 at z=0 and 1 at z = 1}

^{f(z) = F'(z) = 2 - 2z.}