Binomial {base} | R Documentation |

## The Binomial Distribution

### Description

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

and `prob`

.

### 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. |

`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*.

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.

### References

Catherine Loader (2000). *Fast and Accurate Computation of
Binomial Probabilities*; manuscript available from
http://cm.bell-labs.com/cm/ms/departments/sia/catherine/dbinom

### See Also

`dnbinom`

for the negative binomial, and
`dpois`

for the Poisson distribution.

### Examples

# 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)