how to find the compound distribution, if the rate of poisson distribution is normally distributed with mean and variance ? I know I have to find the integral of:
$$ \frac {1} {\sigma \sqrt{2 \pi} x!} \int_0^\infty \lambda^x e^{-{\lambda}} e^{- \frac{(\lambda - \mu)^{2}}{2 \sigma^2}} d \lambda $$
As you can see, the rate of possion distrbution $ \lambda $ is normally distributed with $ \mu$ and $\sigma$
As far as I know, you can do it easier with gamma-poisson distribution. Can anyone point me to the right integration technique ? I need the exact closed form solution of this compound distribution.