Binomial {stats}  R Documentation 
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 
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

Details
The binomial distribution with size
= n
and
prob
= p
has density
p(x) = {n \choose x} {p}^{x} {(1p)}^{nx}
for x = 0, \ldots, 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) \ge 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 saddlepoint expansion is used: see
Catherine Loader (2000). Fast and Accurate Computation of Binomial Probabilities; available as https://www.rproject.org/doc/reports/CLoaderdbinom2002.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)