Search results
Results from the WOW.Com Content Network
Thus, the negative binomial distribution is equivalent to a Poisson distribution with mean pT, where the random variate T is gamma-distributed with shape parameter r and intensity (1 − p). The preceding paragraph follows, because λ = pT is gamma-distributed with shape parameter r and intensity (1 − p )/ p .
Some distributions have been specially named as compounds: beta-binomial distribution, Beta negative binomial distribution, gamma-normal distribution. Examples: If X is a Binomial(n,p) random variable, and parameter p is a random variable with beta(α, β) distribution, then X is distributed as a Beta-Binomial(α,β,n).
Returning to our example, if we pick the Gamma distribution as our prior distribution over the rate of the Poisson distributions, then the posterior predictive is the negative binomial distribution, as can be seen from the table below.
The gamma distribution is also used to model errors in multi-level Poisson regression models because a mixture of Poisson distributions with gamma-distributed rates has a known closed form distribution, called negative binomial.
Negative binomial regression is a popular generalization of Poisson regression because it loosens the highly restrictive assumption that the variance is equal to the mean made by the Poisson model. The traditional negative binomial regression model is based on the Poisson-gamma mixture distribution.
Related to this distribution are a number of other distributions: the displaced Poisson, the hyper-Poisson, the general Poisson binomial and the Poisson type distributions. The Conway–Maxwell–Poisson distribution, a two-parameter extension of the Poisson distribution with an adjustable rate of decay.
The shift geometric distribution is discrete compound Poisson distribution since it is a trivial case of negative binomial distribution. This distribution can model batch arrivals (such as in a bulk queue [5] [9]). The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total ...
The posterior predictive distribution for a single additional observation is a negative binomial distribution, [49]: 53 sometimes called a gamma–Poisson distribution. Simultaneous estimation of multiple Poisson means