Mixed Poisson distribution
A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that the conditional distribution of a random variable, given the value of the rate parameter, is a Poisson distribution, and that the rate parameter itself is considered as a random variable. Hence it is a special case of a compound probability distribution. Mixed Poisson distributions can be found in actuarial mathematics as a general approach for the distribution of the number of claims and is also examined as an epidemiological model. It should not be confused with compound Poisson distribution or compound Poisson process.
Definition
A random variable X satisfies the mixed Poisson distribution with density if it has the probability distributionIf we denote the probabilities of the Poisson distribution by, then
Properties
- The variance is always bigger than the expected value. This property is called overdispersion. This is in contrast to the Poisson distribution where mean and variance are the same.
- In practice, almost only densities of gamma distributions, logarithmic normal distributions and inverse Gaussian distributions are used as densities. If we choose the density of the gamma distribution, we get the negative binomial distribution, which explains why this is also called the Poisson gamma distribution.
Expected value
The expected value of the mixed Poisson distribution isVariance
For the variance one getsSkewness
The skewness can be represented asCharacteristic function
The characteristic function has the formWhere is the moment generating function of the density.