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First of all, I am rather new to statistics, so go easy on me.

I am aware that the negative binomial distribution can be thought to arise as a result of letting the $\lambda$ parameter in a Poisson distribution vary like the Gamma distribution. I wanted to know if there was a similar result for a Poisson-Gaussian mix, where $\lambda$ was a random variable and was distributed as a Gaussian. Obviously there is the issue of Gaussian distributions allowing for negative values, but if we restrict it from zero to infinity and normalize accordingly I thought it should be possible.

In this case, the $\lambda$ parameter would have some average value and a symmetric spread around that value.

From what I understand, the mixture is represented by following integral.

$$\frac{\int_{0}^{\infty}\frac{\lambda^k}{k!}e^{-\lambda} e^{-\frac{(\lambda-\mu)^2}{2 \sigma^2}}d\lambda}{\int_{0}^{\infty}e^{-\frac{(\lambda-\mu)^2}{2 \sigma^2}}d\lambda}$$

If a simple or well-known form does not exist, how would you all recommend implementing a fit for this in a program like Matlab? Computing it from this definition and fitting to it seems rather difficult. I was also looking into a mixing with a log-normal distribution (since $\lambda$ is positive), but I thought that would be even more difficult to deal with.

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    $\begingroup$ On a related note, there is a closed form for the moment generating function Poisson-Inverse Gaussian mixture. See Tremblay, Luc. "Using the Poisson inverse Gaussian in bonus-malus systems." Astin Bulletin 22.1 (1992): 97-106.{PDF} $\endgroup$ – Avraham Jun 19 '14 at 19:39
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    $\begingroup$ How would you go about using that result for fitting, for example? I am not too familiar with fitting with a moment function $\endgroup$ – user157879 Jun 19 '14 at 20:05
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    $\begingroup$ I'm not an expert with it either, but these links may help:(cran.r-project.org/web/packages/gamlss.dist/gamlss.dist.pdf)See PIG) and (jstor.org/discover/10.2307/…) $\endgroup$ – Avraham Jun 19 '14 at 20:06
  • $\begingroup$ Your integral is incorrect: $\lambda$ has the real line as its support and you've only integrated half of it out. You could use the half-normal (the density of $\left| Z \right|$ when $Z \sim N(0, 1)$. You could also use a log-normal. In terms of fitting, I would use Markov Chain Monte Carlo to find the posterior distribution of $\lambda|X=x$. $\endgroup$ – Dennis Jun 20 '14 at 2:16
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    $\begingroup$ $\lambda$ can only have positive values for a Poisson distribution, it would not be normalizable otherwise right? It would also be physically irrelevant, as a Poisson process with $\lambda<0$ doesn't really make sense. Can you also explain your last sentence as well in more detail, possibly in an answer form? $\endgroup$ – user157879 Jun 20 '14 at 15:08

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