# NegHypergeometric: The Negative Hypergeometric Distribution In tolerance: Statistical Tolerance Intervals and Regions

## Description

Density, distribution function, quantile function, and random generation for the negative hypergeometric distribution.

## Usage

 ```1 2 3 4``` ```dnhyper(x, m, n, k, log = FALSE) pnhyper(q, m, n, k, lower.tail = TRUE, log.p = FALSE) qnhyper(p, m, n, k, lower.tail = TRUE, log.p = FALSE) rnhyper(nn, m, n, k) ```

## Arguments

 `x,q` Vector of quantiles representing the number of trials until `k` successes have occurred (e.g., until `k` white balls have been drawn from an urn without replacement). `m` The number of successes in the population (e.g., the number of white balls in the urn). `n` The population size (e.g., the total number of balls in the urn). `k` The number of successes (e.g., white balls) to achieve with the sample. `p` Vector of probabilities, which must be between 0 and 1. `nn` The number of observations. If `length>1`, then the length is taken to be the number required. `log,log.p` Logical vectors. If `TRUE`, then probabilities are given as `log(p)`. `lower.tail` Logical vector. If `TRUE`, then probabilities are P[X≤ x], else P[X>x].

## Details

A negative hypergeometric distribution (sometimes called the inverse hypergeometric distribution) models the total number of trials until `k` successes occur. Compare this to the negative binomial distribution, which models the number of failures that occur until a specified number of successes has been reached. The negative hypergeometric distribution has density

p(x) = choose(x-1, k-1)choose(n-x, m-k) / choose(n, m)

for x=k,k+1,...,n-m+k.

## Value

`dnhyper` gives the density, `pnhyper` gives the distribution function, `qnhyper` gives the quantile function, and `rnhyper` generates random deviates.

Invalid arguments will return value `NaN`, with a warning.

## References

Wilks, S. S. (1963), Mathematical Statistics, Wiley.

## See Also

`runif` and `.Random.seed` about random number generation.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```## Randomly generated data from the negative hypergeometric ## distribution. set.seed(100) x <- rnhyper(nn = 1000, m = 15, n = 40, k = 10) hist(x, main = "Randomly Generated Data", prob = TRUE) x.1 = sort(x) y <- dnhyper(x = x.1, m = 15, n = 40, k = 10) lines(x.1, y, col = 2, lwd = 2) plot(x.1, pnhyper(q = x.1, m = 15, n = 40, k = 10), type = "l", xlab = "x", ylab = "Cumulative Probabilities") qnhyper(p = 0.20, m = 15, n = 40, k = 10, lower.tail = FALSE) qnhyper(p = 0.80, m = 15, n = 40, k = 10) ```

tolerance documentation built on Feb. 6, 2020, 5:08 p.m.