Results 1 to 6 of 6

Math Help - Problem of the week :)

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    12

    Problem of the week :)

    I've been working on this for about 10 days. I know I need some book revising...

    So, there are N+1 r.v.

    N variables (  X_{1},...,X_{N} ) follow a Gamma distribution:  f_{X}(x)=x^{k_{1}-1} \frac {e^{-x}}{\Gamma(k_{1})}, for \text{ }  x>0

    1 variable (  Y ) follows a 'negative' Gamma distribution:  f_{Y}(x)=(-x)^{k_{2}-1} \frac {e^{x}}{\Gamma(k_{2})}, for \text{ } x<0

    I need to find the pdf of the sum:
    Z=\alpha+\frac{1}{N}\sum_{i=1}^{N} \beta_{1} X_i+\beta_{2} Y
    where \alpha,\beta_{1},\beta_{2} - just parameters

    What I did:

    My first method was by characteristic function. I found the ch.f. of the sum and tried to inverse-Fourier-transmorm it, so that it gives the pdf. But I got a cumbersome integral to evaluate (I posted it here: http://www.mathhelpforum.com/math-he...tegration.html ), still no help .

    Convolution method is my second try. I found the pdf of \frac{1}{N}\sum_{i=1}^{N} \beta_{1} X_i and the pdf of \beta_{2} Y

    Now I'm working to find the convolution of these 2 parts. Not very simple (to me), not sure If I can make it.

    Any suggestions to help with my methods or to find the pdf of Z ? Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    how about just using the MGFs?
    I would split it into two parts....

    Z=\alpha+\frac{\beta_1}{N}\sum_{i=1}^{N} X_i+\beta_{2} Y

    Pull out the beta1 and use MGFs on the sum of the gammas and call that W.
    Then write Z as

    Z=\alpha+\frac{\beta_1}{N}W+\beta_{2} Y

    This might not be recognizable as any known distribution, via MGFs
    but you can use Calc 3 to obtain the density here.

    AND you better have independence here.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    12
    Yes, I've split it into two parts then obtained the characteristic function of Z. (I think it's easier to obtain density from CHF than from MGFs). Now, to find the pdf of Z we must evaluate the integral:

    <br />
f_{Z}(x)=\int^{\infty}_{-\infty} \frac{e^{-i u (x-\alpha)}}{(1+i u \beta_{1})^{k_{1}}(1-i u \frac{\beta_{2}}{N})^{N k_{2}}}du<br />

    the density will then consist of 2 parts: for  x>\alpha :  \int^{\infty}_{0} and for x<\alpha :  \int^{0}_{-\infty}

    From Calculus 3, any suggestion to solve this integral ? I'm having some trouble with that. Thank you the Beagle.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1
    Quote Originally Posted by stokastik View Post
    any suggestion to solve this integral ? I'm having some trouble with that.
    I mean you could just use Wolfram.......
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Apr 2010
    Posts
    12
    I've tried but Wolfram and Maple couldn't do it, so...
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Apr 2010
    Posts
    12
    Finally I made it!
    I found the pdf...

    thanks everyone
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 18th 2009, 01:28 PM
  2. Problem Of The Week
    Posted in the Math Topics Forum
    Replies: 4
    Last Post: March 26th 2007, 06:58 AM

Search Tags


/mathhelpforum @mathhelpforum