| 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 (non-negative) means. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are
|
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.
rpois returns a vector of type integer unless generated
values exceed the maximum representable integer when double
values are returned.
Source
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.
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")
## The (limit) case lambda = 0 :
stopifnot(identical(dpois(0,0), 1),
identical(ppois(0,0), 1),
identical(qpois(1,0), 0))