stats (version 3.4.3)

# 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.

## 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
``````# NOT RUN {
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.