Results 1 to 10 of 10
Like Tree1Thanks
  • 1 Post By chiro

Math Help - finding expectation

  1. #1
    Junior Member
    Joined
    Dec 2011
    Posts
    57
    Thanks
    1

    finding expectation

    The joint density function of X and Y is given by
    f(x, y) = \frac{1}{2\sqrt{2\pi}}\frac{e^{-y/2}}{y^{3/2}}, where |x| < y < \infty .

    Find E[X|Y].


    How do I go about doing this question? Any help is appreciated. Thanks in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,698
    Thanks
    620

    Re: finding expectation

    Hey alphabeta89.

    To start off with: what did you get for your region of integration?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2011
    Posts
    57
    Thanks
    1

    Re: finding expectation

    Quote Originally Posted by chiro View Post
    Hey alphabeta89.

    To start off with: what did you get for your region of integration?
    -y<x<y, 0<y<\infty
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,698
    Thanks
    620

    Re: finding expectation

    Hint: Integrate out the x variable first and use the properties of the Gamma Function:

    Gamma function - Wikipedia, the free encyclopedia
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Dec 2011
    Posts
    57
    Thanks
    1

    Re: finding expectation

    Do we need to find out what is f_{X|Y}{(x|y)}? Then use E[X|Y] = \int_{-\infty}^{\infty} {x f_{X|Y}{(x|y)}}\mathrm{d}x?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,698
    Thanks
    620

    Re: finding expectation

    You could find the conditional distribution or you could just condition Y on the appropriate values and get the expectation.

    They will both give you the same result and the rigorous way to prove it is to use Fubini's Theorem.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Dec 2011
    Posts
    57
    Thanks
    1

    Re: finding expectation

    Hi, just to check, is this correct?

    f_Y{(y)}=\int_{-y}^{y}K\frac{e^{-y/2}}{y^{3/2}}\mathrm{d}x= K\frac{2e^{-y/2}}{\sqrt{y}}=\frac{e^{-y/2}}{\sqrt{2\pi{y}}}

    f_{X|Y}{(x|y)}=\frac{f_{X,Y}{(x,y)}}{f_Y{(y)}}= \frac{\left (\frac{1}{2\sqrt{2\pi}}\frac{e^{-y/2}}{y^{3/2}}\right )}{\left (\frac{e^{-y/2}}{\sqrt{2\pi{y}}}  \right )}=\frac{1}{2y}

    E{[X|Y]}&=\int_{-\infty}^{\infty} {x f_{X|Y}{(x|y)}}\mathrm{d}x=\int_{-y}^{y}{x\cdot{\frac{1}{2y}}}\mathrm{d}x=0
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,698
    Thanks
    620

    Re: finding expectation

    Can you show the step by step integration techniques to get f_y(Y)?

    The steps and approach look good.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Dec 2011
    Posts
    57
    Thanks
    1

    Re: finding expectation

    Quote Originally Posted by chiro View Post
    Can you show the step by step integration techniques to get f_y(Y)?

    The steps and approach look good.
    f_Y{(y)}=\int_{-\infty}^{\infty}f_{X,Y}{(x,y)}\mathrm{d}x=\int_{-y}^{y}\frac{1}{2\sqrt{2\pi}}\frac{e^{-y/2}}{y^{3/2}}\mathrm{d}x= \frac{1}{2\sqrt{2\pi}}\frac{2e^{-y/2}}{\sqrt{y}}=\frac{e^{-y/2}}{\sqrt{2\pi{y}}}.

    Is this correct?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,698
    Thanks
    620

    Re: finding expectation

    That looks right algebraically.

    One way to confirm the theoretical results is to simulate the random variables in a statistical package if you so choose to do.
    Thanks from alphabeta89
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding the expectation of X_2
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: August 25th 2011, 07:22 AM
  2. Finding expectation through conditioning
    Posted in the Statistics Forum
    Replies: 2
    Last Post: July 27th 2011, 09:45 PM
  3. Replies: 0
    Last Post: May 4th 2010, 08:38 AM
  4. plz help in finding Expectation and Covariance
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 27th 2010, 11:55 PM
  5. Finding the expectation of a constant bias
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: April 10th 2009, 11:04 PM

Search Tags


/mathhelpforum @mathhelpforum