# Difficult integral (product of gamma func with exp)

1. Dec 8, 2009

### coolnessitself

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

$$n,\theta$$ 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 $$P(X<2)$$ where $$X_i\sim \mathrm{Exp}(\theta)$$. The MLE of an exp is $$\overline{X}$$, and this is invariant under transforms, and so we need to solve the expectation of $$1-\exp\left\{\frac{-2n}{\overline{x}}\right\}$$. So the 1 is easy, and since here $$\overline{X}\sim\Gamma$$, it's the expectation of $$\frac{-2n}{x}\right\}$$ 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