- #1
Gekko
- 71
- 0
Homework Statement
Variables X and Y are uniformly distributed on [-1,1]
Z = X^2 + Y^2 where X^2+Y^2 <= 1
Show that Z is uniformly distributed on [0,1]
The Attempt at a Solution
If we set X=Rcos(theta) and Y=Rsin(theta), the joint pdf is R/pi where 0<=R<=1 and 0<=theta<=2pi
So, since the interval [0,1] is just the top right quadrant of the circle, we can integrate R/pi where 0<=R<=1 and 0<=theta<=pi/2 which gives 1/4
Does this prove the uniform distribution? My question is what is the correct mathematical approach to best prove that Z is uniformly distributed?