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The discussion here presents two relationships between the gamma distribution and the Poisson distribution. In the first one, mixing conditional Poisson distributions with gamma mixing weights produces a negative binomial distribution. Taking the shape parameter and rate parameter in (1) produces the negative binomial distribution as in (2).
Is the gamma Poisson model subject to poor accuracy?
The Gamma distribution is a very flexible distribution which may take on varying shapes, hence it can be expected that for small sample sizes the parameters of the Gamma distribution will be subject to poor accuracy. In this study, we investigated the Gamma–Poisson model using a simulation study.
How are the shape and rate of gamma distributions related?
Taking the shape parameter and rate parameter in (1) produces the negative binomial distribution as in (2). This is one interpretation of the two gamma parameters. In the second interpretation, the rate parameter of the gamma model is intimately tied to the rate parameter of a Poisson process.
When does a conditional random variable have a Poisson distribution?
When a conditional random variable has a Poisson distribution such that its mean is an unknown random quantity but follows a gamma distribution with parameters and as described in (1), the unconditional distribution for has a negative binomial distribution as described in (2).
What is the compound Poisson distribution for a portfolio?
For each portfolio the aggregate claims variable has a compound Poisson distribution. For one of the portfolios, the Poisson parameter is and the individual claim amount has an exponential distribution with parameter . The corresponding Poisson and exponential parameters for the other portfolio are and , respectively.
Is the MGF of a compound Poisson distribution?
The mgf of has the form of a compound Poisson distribution where the Poisson parameter is. Note that the component in the exponent is the mgf of the claim amount distribution. Since it is the weighted average of the individual claim amount mgf’s, this indicates that the distribution function of is the mixture of the distribution functions.