| 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
|
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.
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 \le 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")