GammaPoiss {extraDistr}

Gamma-Poisson distribution

Description

Probability mass function and random generation for the gamma-Poisson distribution.

Usage

dgpois(x, shape, rate, scale = 1/rate, log = FALSE)

pgpois(q, shape, rate, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)

rgpois(n, shape, rate, scale = 1/rate)

Arguments

x, q

vector of quantiles.

shape, scale

shape and scale parameters. Must be positive, scale strictly.

rate

an alternative way to specify the scale.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X \le x] otherwise, P[X > x].

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Gamma-Poisson distribution arises as a continuous mixture of Poisson distributions, where the mixing distribution of the Poisson rate \lambda is a gamma distribution. When X \sim \mathrm{Poisson}(\lambda) and \lambda \sim \mathrm{Gamma}(\alpha, \beta), then X \sim \mathrm{GammaPoisson}(\alpha, \beta).

Probability mass function

f(x) = \frac{\Gamma(\alpha+x)}{x! \, \Gamma(\alpha)} \left(\frac{\beta}{1+\beta}\right)^x \left(1-\frac{\beta}{1+\beta}\right)^\alpha

Cumulative distribution function is calculated using recursive algorithm that employs the fact that \Gamma(x) = (x - 1)!. This enables re-writing probability mass function as

f(x) = \frac{(\alpha+x-1)!}{x! \, \Gamma(\alpha)} \left( \frac{\beta}{1+\beta} \right)^x \left( 1- \frac{\beta}{1+\beta} \right)^\alpha

what makes recursive updating from x to x+1 easy using the properties of factorials

f(x+1) = \frac{(\alpha+x-1)! \, (\alpha+x)}{x! \,(x+1) \, \Gamma(\alpha)} \left( \frac{\beta}{1+\beta} \right)^x \left( \frac{\beta}{1+\beta} \right) \left( 1- \frac{\beta}{1+\beta} \right)^\alpha

and let's us efficiently calculate cumulative distribution function as a sum of probability mass functions

F(x) = \sum_{k=0}^x f(k)

See Also

Gamma, Poisson

Examples

x <- rgpois(1e5, 7, 0.002)
xx <- seq(0, 12000, by = 1)
hist(x, 100, freq = FALSE)
lines(xx, dgpois(xx, 7, 0.002), col = "red")
hist(pgpois(x, 7, 0.002))
xx <- seq(0, 12000, by = 0.1)
plot(ecdf(x))
lines(xx, pgpois(xx, 7, 0.002), col = "red", lwd = 2)