Results 1 to 2 of 2

Math Help - find pdf of z= x^2+y^2, x,y-uniform

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    4

    find pdf of z= x^2+y^2, x,y-uniform

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    I don't really understand your computation, but here is what I'd say:

    Initially, the point (X,Y) is uniformly distributed on the square [-1,1]\times [-1,1]. Discarding the samples where Z>1 amounts to sample the distribution of (X,Y) conditional on the event that Z\leq 1, which is the uniform distribution on the unit disc D(0,1) (we discard the points falling outside).
    Then, for 0<r<1: P(Z\leq r)=P((X,Y)\in D(0,\sqrt{r}))=\frac{\mbox{Area}(D(0,\sqrt{r}))}{\  mbox{Area}(D(0,1))}=\frac{\pi r}{\pi}=r. This proves that the distribution of Z is the uniform distribution on [0,1].
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 31st 2010, 07:09 PM
  2. if X is uniform, find density function of Y=X^2
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 8th 2010, 02:36 PM
  3. The uniform and its log
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: January 27th 2010, 07:00 AM
  4. uniform differentiable => uniform continuity
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 30th 2009, 03:19 PM
  5. Uniform Continuous and Uniform Convergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 28th 2007, 02:51 PM

Search Tags


/mathhelpforum @mathhelpforum