R: The Poisson Distribution

Poisson {base} | 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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