Poisson: The Poisson Distribution

PoissonR Documentation

The Poisson Distribution


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


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)



vector of (non-negative integer) quantiles.


vector of quantiles.


vector of probabilities.


number of random values to return.


vector of (non-negative) means.

log, log.p

logical; if TRUE, probabilities p are given as log(p).


logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].


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.

Note that λ = 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 ≤ 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.


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.

rpois returns a vector of type integer unless generated values exceed the maximum representable integer when double values are returned since R version 4.0.0.


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

Distributions for other standard distributions, including dbinom for the binomial and dnbinom for the negative binomial distribution.




-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))