Binomial The Binomial Distribution

Description

Density, distribution function, quantile function and random generation for the binomial distribution with parameters size and prob.

This is conventionally interpreted as the number of ‘successes’ in size trials.

Usage

dbinom(x, size, prob, log = FALSE)
pbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)
qbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE)
rbinom(n, size, prob)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
size	number of trials (zero or more).
prob	probability of success on each trial.
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 binomial distribution with size = n and prob = p has density

p(x) = choose(n, x) p^x (1-p)^(n-x)

for x = 0, …, n. Note that binomial coefficients can be computed by choose in R.

If an element of x is not integer, the result of dbinom is zero, with a warning.

p(x) is computed using Loader's algorithm, see the reference below.

The quantile is defined as the smallest value x such that F(x) ≥ p, where F is the distribution function.

Value

dbinom gives the density, pbinom gives the distribution function, qbinom gives the quantile function and rbinom generates random deviates.

If size is not an integer, NaN is returned.

The length of the result is determined by n for rbinom, 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.

Source

For dbinom a saddle-point expansion is used: see

Catherine Loader (2000). Fast and Accurate Computation of Binomial Probabilities; available as https://www.r-project.org/doc/reports/CLoader-dbinom-2002.pdf

pbinom uses pbeta.

qbinom uses the Cornish–Fisher Expansion to include a skewness correction to a normal approximation, followed by a search.

rbinom (for size < .Machine$integer.max) is based on

Kachitvichyanukul, V. and Schmeiser, B. W. (1988) Binomial random variate generation. Communications of the ACM, 31, 216–222.

For larger values it uses inversion.

See Also

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

Examples

require(graphics)
# Compute P(45 < X < 55) for X Binomial(100,0.5)
sum(dbinom(46:54, 100, 0.5))

## Using "log = TRUE" for an extended range :
n <- 2000
k <- seq(0, n, by = 20)
plot (k, dbinom(k, n, pi/10, log = TRUE), type = "l", ylab = "log density",
      main = "dbinom(*, log=TRUE) is better than  log(dbinom(*))")
lines(k, log(dbinom(k, n, pi/10)), col = "red", lwd = 2)
## extreme points are omitted since dbinom gives 0.
mtext("dbinom(k, log=TRUE)", adj = 0)
mtext("extended range", adj = 0, line = -1, font = 4)
mtext("log(dbinom(k))", col = "red", adj = 1)