Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Difficult integral (product of gamma func with exp)

  1. Dec 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Solve. [tex]\int\limits_{0}^\infty \exp\left\{\frac{-2n}{x} - \frac{x}{\theta}\right\} x^{n-1} dx[/tex]

    [tex]n,\theta[/tex] are constants.
    2. Relevant equations

    3. The attempt at a solution
    So actually this problem is from the realm of stats...find the MVUE of [tex]P(X<2)[/tex] where [tex]X_i\sim \mathrm{Exp}(\theta)[/tex]. The MLE of an exp is [tex]\overline{X}[/tex], and this is invariant under transforms, and so we need to solve the expectation of [tex]1-\exp\left\{\frac{-2n}{\overline{x}}\right\}[/tex]. So the 1 is easy, and since here [tex]\overline{X}\sim\Gamma[/tex], it's the expectation of [tex]\frac{-2n}{x}\right\}[/tex] in a gamma distribution, resulting in the integral above, with some constants I've removed. The exam is over, no one could find a solution. Can the integral be solved? I tried a u sub for sqrt(y..) and tried intrgrating by parts n-1 times, and a few other ideas. Any ideas for a method to solve the above integral?
    Last edited: Dec 8, 2009
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?