stats (version 3.3)

# 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 \le x]\$, otherwise, \$P[X > x]\$.

## 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 from http://www.herine.net/stat/software/dbinom.html.

`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.

## Details

The binomial distribution with `size` \$= n\$ and `prob` \$= p\$ has density \$\$p(x) = {n \choose x} {p}^{x} {(1-p)}^{n-x}\$\$ 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.

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

## Examples

Run this code
``````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.