Poisson: The Poisson Distribution

Description

Density, distribution function, quantile function and random generation for the Poisson distribution with parameter lambda.

Usage

dpois(x, lambda, log = FALSE)
ppois(q, lambda, lower.tail = TRUE, log.p = FALSE)
qpois(p, lambda, lower.tail = TRUE, log.p = FALSE)
rpois(n, lambda)

Arguments

vector of (non-negative integer) quantiles.

vector of quantiles.

vector of probabilities.

number of random values to return.

lambda

vector of (non-negative) means.

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, $P[X > x]$.

Value

dpois gives the (log) density, ppois gives the (log) distribution function, qpois gives the quantile function, and rpois generates random deviates.Invalid lambda will result in return value NaN, with a warning.The length of the result is determined by n for rpois, and is the maximum of the lengths of the numerical arguments for the other functions.The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Source

dpois uses C code contributed by Catherine Loader (see dbinom). ppois uses pgamma. qpois uses the Cornish--Fisher Expansion to include a skewness correction to a normal approximation, followed by a search. rpois uses Ahrens, J. H. and Dieter, U. (1982). Computer generation of Poisson deviates from modified normal distributions. ACM Transactions on Mathematical Software, 8, 163--179.

Details

The Poisson distribution has density

$p (x) = \frac{λ^{x} e^{- λ}}{x!}$ for $x = 0, 1, 2, \ldots$ . The mean and variance are $E(X) = Var(X) = \lambda$.

If an element of x is not integer, the result of dpois is zero, with a warning. $p(x)$ is computed using Loader's algorithm, see the reference in dbinom.

The quantile is right continuous: qpois(p, lambda) is the smallest integer $x$ such that $P(X \le x) \ge p$.

Setting lower.tail = FALSE allows to get much more precise results when the default, lower.tail = TRUE would return 1, see the example below.

Examples

Run this code

require(graphics)

-log(dpois(0:7, lambda = 1) * gamma(1+ 0:7)) # == 1
Ni <- rpois(50, lambda = 4); table(factor(Ni, 0:max(Ni)))

1 - ppois(10*(15:25), lambda = 100)  # becomes 0 (cancellation)
    ppois(10*(15:25), lambda = 100, lower.tail = FALSE)  # no cancellation

par(mfrow = c(2, 1))
x <- seq(-0.01, 5, 0.01)
plot(x, ppois(x, 1), type = "s", ylab = "F(x)", main = "Poisson(1) CDF")
plot(x, pbinom(x, 100, 0.01), type = "s", ylab = "F(x)",
     main = "Binomial(100, 0.01) CDF")

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