Determine Joint Density & E[z] of f_xy(x,y) Function

[cutesteph]

Homework Statement

f_xy(x,y)= 24xy for o<x<1 and 0<y<1-x

let z=[1-x]/y and w=y

determine joint density of wz
and E(z)

Homework Equations

The Attempt at a Solution

E[z] = Integral [0,1] integral [0,1-x] 24xy*(1-x)/y dydx = 2

The joint distribution doing a transformation to x=1-zy and y =w so x = 1-wz

Jacobian = -w

so f_wz (wz) = 24(1-zw)w *|-w| = 24 (1-zw)w^2 the new bounds are 0<1-zw<1 => 1>zw>0 and 0<w<1-(1-zw) => 0<w<wz

the bounds are 0<w<1 and o<z<1 but the density is not equaling 1 so I am doing something wrong.

 

[haruspex]

I don't understand how you got from:

the new bounds are 0<1-zw<1 => 1>zw>0 and 0<w<1-(1-zw) => 0<w<wz

which is right, to

the bounds are 0<w<1 and o<z<1

 

[cutesteph]

So we have 1>zw>0 and zw>w>0

So 1>zw>w>0 => 1/w > z > 1 and it seems 1>w>0 but the integration does not work.

 

[haruspex]

So we have 1>zw>0 and zw>w>0

So 1>zw>w>0 => 1/w > z > 1 and it seems 1>w>0 but the integration does not work.

Your expression for joint pdf integrates to 1 for me. Please show your working,