Results 1 to 5 of 5

Math Help - Joint Density function Question

  1. #1
    Junior Member
    Joined
    Aug 2010
    Posts
    53

    Joint Density function Question

    Hi all,

    I have been going though a Q and have no idea how to even start it:

    Q:

    Suppose that X,Y have the joint distribution:

    f(x,y) = c(1-x^2-y^2)^{1/2} , x^2 + y^2 \le 1

    Find c..?

    Now my initial thoughts are that the bounded unit cirrcle region gives the limits of integration such that:

    P(X,Y \subset A)= _A\int \int f(x,y) dydx, in our case A is bound by the unit circle, giving:

    = c \displaystyle \int_{-(1-x^2)^{1/2}}^{(1-x^2)^{1/2}} \int_{-(1-y^2)^{1/2}}^{(1-y^2)^{1/2}} 1-x^2-y^2 dx dy

    does this look like the right method to employ? I am not convinced..... the answer in the text book (no method though) 3/2 \pi... can anyone offer any guidance on this one??

    Many thanks for reading!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Dec 2010
    Posts
    470
    Try to integrate in polar coordiantes.

    r^2 = x^2 + y^2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    yes you need polar, but your bounds and integrand are wrong in rectangular as well
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    It would be

    1= c \displaystyle \int_{-1}^1\int_{-(1-y^2)^{1/2}}^{(1-y^2)^{1/2}} \sqrt{1-x^2-y^2} dx dy

    which becomes, in polar....

    1= c \displaystyle \int_0^{2\pi}\int_0^1 r\sqrt{1-r^2}dr d\theta
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Aug 2010
    Posts
    53
    thats great, thanks a lot guys.

    just seen my mistake... dont worry!
    Last edited by mr fantastic; January 12th 2011 at 07:36 PM. Reason: Merged posts.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Need help with a joint density function
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: February 19th 2010, 07:58 AM
  2. Replies: 5
    Last Post: December 6th 2009, 12:30 AM
  3. Joint density function
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: November 19th 2009, 07:50 AM
  4. About the joint density function
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: October 20th 2008, 12:33 PM
  5. Joint density function
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: April 9th 2008, 10:52 AM

Search Tags


/mathhelpforum @mathhelpforum