# Binomial

##### The Binomial Distribution

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.

- Keywords
- distribution

##### 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, \(P[X > x]\).

##### 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.

##### 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.

##### See Also

Distributions for other standard distributions, including
`dnbinom`

for the negative binomial, and
`dpois`

for the Poisson distribution.

##### Examples

`library(stats)`

```
# 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.
mtext("dbinom(k, log=TRUE)", adj = 0)
mtext("extended range", adj = 0, line = -1, font = 4)
mtext("log(dbinom(k))", col = "red", adj = 1)
# }
```

*Documentation reproduced from package stats, version 3.6.1, License: Part of R 3.6.1*