# NegBinomial: The Negative Binomial Distribution

## Description

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

## Usage

 ```1 2 3 4``` ```dnbinom(x, size, prob, mu, log = FALSE) pnbinom(q, size, prob, mu, lower.tail = TRUE, log.p = FALSE) qnbinom(p, size, prob, mu, lower.tail = TRUE, log.p = FALSE) rnbinom(n, size, prob, mu) ```

## Arguments

 `x` vector of (non-negative integer) quantiles. `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` target for number of successful trials, or dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer. `prob` probability of success in each trial. `0 < prob <= 1`. `mu` alternative parametrization via mean: see ‘Details’. `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

Γ(x+n)/(Γ(n) x!) p^n (1-p)^x

for x = 0, 1, 2, …, n > 0 and 0 < p ≤ 1.

This represents the number of failures which occur in a sequence of Bernoulli trials before a target number of successes is reached. The mean is μ = n(1-p)/p and variance n(1-p)/p^2.

A negative binomial distribution can also arise as a mixture of Poisson distributions with mean distributed as a gamma distribution (see `pgamma`) with scale parameter `(1 - prob)/prob` and shape parameter `size`. (This definition allows non-integer values of `size`.)

An alternative parametrization (often used in ecology) is by the mean `mu` (see above), and `size`, the dispersion parameter, where `prob` = `size/(size+mu)`. The variance is `mu + mu^2/size` in this parametrization.

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

The case `size == 0` is the distribution concentrated at zero. This is the limiting distribution for `size` approaching zero, even if `mu` rather than `prob` is held constant. Notice though, that the mean of the limit distribution is 0, whatever the value of `mu`.

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.

Invalid `size` or `prob` will result in return value `NaN`, with a warning.

The length of the result is determined by `n` for `rnbinom`, 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.

`rnbinom` returns a vector of type integer unless generated values exceed the maximum representable integer when `double` values are returned since R version 4.0.0.

## Source

`dnbinom` computes via binomial probabilities, using code contributed by Catherine Loader (see `dbinom`).

`pnbinom` uses `pbeta`.

`qnbinom` uses the Cornish–Fisher Expansion to include a skewness correction to a normal approximation, followed by a search.

`rnbinom` uses the derivation as a gamma mixture of Poissons, see

Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer-Verlag, New York. Page 480.

Distributions for standard distributions, including `dbinom` for the binomial, `dpois` for the Poisson and `dgeom` for the geometric distribution, which is a special case of the negative binomial.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28``` ```require(graphics) 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, prob = 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) ## Alternative parametrization x1 <- rnbinom(500, mu = 4, size = 1) x2 <- rnbinom(500, mu = 4, size = 10) x3 <- rnbinom(500, mu = 4, size = 100) h1 <- hist(x1, breaks = 20, plot = FALSE) h2 <- hist(x2, breaks = h1\$breaks, plot = FALSE) h3 <- hist(x3, breaks = h1\$breaks, plot = FALSE) barplot(rbind(h1\$counts, h2\$counts, h3\$counts), beside = TRUE, col = c("red","blue","cyan"), names.arg = round(h1\$breaks[-length(h1\$breaks)])) ```