Poisson {stats}

R Documentation

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

x	vector of (non-negative integer) quantiles.
q	vector of quantiles.
p	vector of probabilities.
n	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].

Details

The Poisson distribution has density

p(x) = \frac{\lambda^x e^{-\lambda}}{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 left continuous: qpois(q, lambda) is the largest integer x such that P(X \le x) < q.

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

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.

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.

See Also

dbinom for the binomial and dnbinom for the negative binomial distribution.

Examples

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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