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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
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