GammaPoiss function

Gamma-Poisson distribution

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

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 $X ~ Poisson(\lambda)$

and $\lambda ~ Gamma(\alpha, \beta)$, then $X ~ Gamma-Poisson(\alpha, \beta)$.

Probability mass function

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

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

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

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)

See Also

Gamma, Poisson