NegBinomial {base} | R Documentation

## The Negative Binomial Distribution

### Description

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

### Usage

    dnbinom(x, size, prob, log = FALSE)
    pnbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)
    qnbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE)
    rnbinom(n, size, prob)

### Arguments

`x, q` | vector of quantiles representing the number of failures which occur in a sequence of Bernoulli trials before a target number of successes is reached, or alternately the probability distribution of a compound Poisson process whose intensity is distributed as a gamma (`pgamma`) distribution with scale parameter `(1-prob)/prob` and shape parameter `size` (this definition allows non-integer values of `size`).
`x` | vector of (non-negative integer) quantiles.
`q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations to generate.
`size` | target for number of successful trials, or shape parameter of gamma distribution.
`prob` | probability of success in each trial, or scale of gamma distribution (`prob` = `scale/(1+scale)`).
`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 negative binomial distribution with `size` = n and `prob` = p has density

p(x) = Gamma(x+n)/(Gamma(n) x!) p^n (1-p)^x

for x = 0, 1, 2, ...

If an element of `x` is not integer, the result of `dnbinom` is zero, with a warning.

The quantile is defined as the smallest value x such that F(x) >= p, where F is the distribution function.

### Value

`dnbinom` gives the density, `pnbinom` gives the distribution function, `qnbinom` gives the quantile function, and `rnbinom` generates random deviates.

### See Also

`dbinom` for the binomial, `dpois` for the Poisson and `dgeom` for the geometric distribution, which is a special case of the negative binomial.

### Examples

    x <- 0:11
    dnbinom(x, size = 1, prob = 1/2) * 2^(1 + x) # == 1
    126 /  dnbinom(0:8, size  = 2, prob  = 1/2) #- theoretically integer

    ## Cumulative ('p') = Sum of discrete prob.s ('d');  Relative error :
    summary(1 - cumsum(dnbinom(x, size = 2, prob = 1/2)) /
                      pnbinom(x, size  = 2, prob = 1/2))

    x <- 0:15
    size <- (1:20)/4
    persp(x,size, dnb <- outer(x,size,function(x,s)dnbinom(x,s, pr= 0.4)),
          xlab = "x", ylab = "s", zlab="density", theta = 150)
    title(tit <- "negative binomial density(x,s, pr = 0.4)  vs.  x & s")

    image  (x,size, log10(dnb), main= paste("log [",tit,"]"))
    contour(x,size, log10(dnb),add=TRUE)