Results 1 to 6 of 6

Math Help - Distribution of a + X

  1. #1
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1

    Distribution of a + X

    Suppose that Θ is a random variable that follows a gamma distribution with parameters λ and α, where α is an integer, and suppose that, conditional on Θ, Χ follows a poisson distribution with parameter Θ. Find the unconditional distribution of α + Χ. (Hint: Find the mgf by using iterated conditional expectations)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    Use the hint.

     E(e^{(a+X)t}) =E(E(e^{(a+X)t})|\theta)

     =e^{at}E(E(e^{tX})|\theta) =e^{at}E(e^{\theta(e^t-1)})

     =e^{at}M_{\theta}(e^t-1)=e^{at}M_{\theta}(s)

    SO, plug in  s=e^t-1 into the MGF of your gamma distribution.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    Did you get an interesting result?
    I didn't finish this, because I think that's your job.
    But I would like to see what your final result is.
    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
    I'll post the result when I'm finished.

    One interesting thing I've found:

    X \sim  Negative Binomial(\alpha, \frac{1}{1+\lambda})
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1
    I just did this in the past fifteen minutes so please let me know if things go awry.

    M_{a+X} = E[e^{(a+X)t}] = e^{at}M_{\theta} (s) = e^{at} (\frac{\lambda}{\lambda -(e^t - 1)})^a

    Now,

    M'_{a+X} (0) = a(\frac{\lambda + a}{\lambda}) = E[a+X]

    Also,

    E[a+X] = \int_{-\infty}^{\infty} (a+x)f_{a+X} (x) dx

    Therefore,

    \int_{-\infty}^{\infty} (a+x)f_{a+X} (x) dx = a(\frac{\lambda + 1}{\lambda})

    And I'm stuck here. Can I take the derivative of both sides or something?

    f_{a+x} (x) = \frac{1}{(a+x)}
    Last edited by Anonymous1; February 7th 2010 at 06:20 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1
    \alpha + X \sim Negative Binomial
    Last edited by Anonymous1; February 8th 2010 at 08:41 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: May 12th 2011, 03:43 PM
  2. normal distribution prior and posterior distribution proof
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: March 9th 2011, 06:12 PM
  3. Replies: 2
    Last Post: March 29th 2010, 02:05 PM
  4. Conditional distribution and finding unconditional distribution
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: December 2nd 2008, 04:48 AM
  5. Cumulative distribution function of binomial distribution
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: October 31st 2008, 03:34 PM

Search Tags


/mathhelpforum @mathhelpforum