f(x,y)=12xy(1−y), x,y∈(0,1). Find distribution of Z=XY^2

recycle4 · Dec 5, 2013

Consider two random variables X and Y with the joint probability density:

12xy(1-y) for 0<x<1, 0<y<1
0 elsewhere.

Find the probability density of Z = XY^2 to determine the joint probability density of Y and Z and then integrating out y.

HallsofIvy · Dec 5, 2013

recycle4 said:
Consider two random variables X and Y with the joint probability density:

12xy(1-y) for 0<x<1, 0<y<1
0 elsewhere.
Find the probability density of Z = XY^2 to determine the joint probability density of Y and Z and then integrating out y.

Do you not know the basic definitions? If Z= XY^2 then \(\displaystyle X= Z/Y^2\). Now replace x with \(\displaystyle z/y^2\)
in 12xy(1- y).

Then integrate that result, with respect to y, from 0 to 1.

f(x,y)=12xy(1−y), x,y∈(0,1). Find distribution of Z=XY^2

recycle4

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HallsofIvy

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