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 positive means.
log, log.p logical; if TRUE, probabilities p are given as log(p).
lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

Details

The Poisson distribution has density

p(x) = lambda^x exp(-lambda)/x!

for x = 0, 1, 2, .... The mean and variance are E(X) = Var(X) = λ.

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: qgeom(q, prob) is the largest integer x such that P(X <= 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.

See Also

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

Examples

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

1 - ppois(10*(15:25), lambda=100)               # becomes 0 (cancellation)
    ppois(10*(15:25), lambda=100, lower=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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