# Poisson {stats}

### 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 ≤ x], otherwise, P[X > x].

### Details

The Poisson distribution has density

p(x) = λ^x exp(-λ)/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 right continuous: `qpois(p, lambda)`

is the smallest integer x such that P(X ≤ x) ≥ 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.

### Values

`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 parameters for the other functions. The numerical parameters other than `n`

are recycled to the length of the result. Only the first elements of the logical parameters are used.

### See Also

Distributions for other standard distributions, including `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")

Documentation reproduced from R 3.0.2. License: GPL-2.