R: The Binomial Distribution

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.

### 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
    plot (0:n,     dbinom(0:n, n, pi/10, log=TRUE), type='l',
          main = "dbinom(*, log=TRUE) is better than  log(dbinom(*))")
    lines(0:n, log(dbinom(0:n, n, pi/10)), col='red', lwd=2)
    mtext("dbinom(k, log=TRUE)", adj=0)
    mtext("extended range", adj=0, line = -1, font=4)
    mtext("log(dbinom(k))", col="red", adj=1)

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