X,Y are independent and uniformly distributed on [-1,1]. Z is constructed by taking the samles of X and Y and constructing X^2+Y^2 , but discarding all Z>1. Therefore, prove taht Z has a uniform distribution on [0,1].

Solution:

So i've substittuted x=r cos a y= r sin a. Then I find that J=1 , => f(r)= r* (Pi/2). Then to find f(Z) = f(R^2):

P(0<R^2<r)= P(R< r^1/2)=F(r^1/2) => f(R^2) = 1/2& r^(-1/2)*f(r^1/2) =>

f(R^1/2) = 1/4*Pi But as I understand the Integral of this function over [0,1] should be 1 and it is not. So do I miss some boundary conditions?

Thanks.