# Poisson

##### The Poisson Distribution

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

.

- Keywords
- distribution

##### 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 \(E(X) = Var(X) = \lambda\).

Note that \(\lambda = 0\) is really a limit case (setting \(0^0 = 1\)) resulting in a point mass at \(0\), see also the example.

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.

##### 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.

##### See Also

Distributions for other standard distributions, including
`dbinom`

for the binomial and `dnbinom`

for
the negative binomial distribution.

##### Examples

`library(stats)`

```
# NOT RUN {
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")
## The (limit) case lambda = 0 :
stopifnot(identical(dpois(0,0), 1),
identical(ppois(0,0), 1),
identical(qpois(1,0), 0))
# }
```

*Documentation reproduced from package stats, version 3.6.2, License: Part of R 3.6.2*