# NAG Library Function Document

## 1Purpose

nag_hypergeom_dist (g01blc) returns the lower tail, upper tail and point probabilities associated with a hypergeometric distribution.

## 2Specification

 #include #include
 void nag_hypergeom_dist (Integer n, Integer l, Integer m, Integer k, double *plek, double *pgtk, double *peqk, NagError *fail)

## 3Description

Let $X$ denote a random variable having a hypergeometric distribution with parameters $n$, $l$ and $m$ ($n\ge l\ge 0$, $n\ge m\ge 0$). Then
 $ProbX=k= m k n-m l-k n l ,$
where $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(0,l-\left(n-m\right)\right)\le k\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(l,m\right)$, $0\le l\le n$ and $0\le m\le n$.
The hypergeometric distribution may arise if in a population of size $n$ a number $m$ are marked. From this population a sample of size $l$ is drawn and of these $k$ are observed to be marked.
The mean of the distribution $\text{}=\frac{lm}{n}$, and the variance $\text{}=\frac{lm\left(n-l\right)\left(n-m\right)}{{n}^{2}\left(n-1\right)}$.
nag_hypergeom_dist (g01blc) computes for given $n$, $l$, $m$ and $k$ the probabilities:
 $plek=ProbX≤k pgtk=ProbX>k peqk=ProbX=k .$
The method is similar to the method for the Poisson distribution described in Knüsel (1986).

## 4References

Knüsel L (1986) Computation of the chi-square and Poisson distribution SIAM J. Sci. Statist. Comput. 7 1022–1036

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: the parameter $n$ of the hypergeometric distribution.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{l}$IntegerInput
On entry: the parameter $l$ of the hypergeometric distribution.
Constraint: $0\le {\mathbf{l}}\le {\mathbf{n}}$.
3:    $\mathbf{m}$IntegerInput
On entry: the parameter $m$ of the hypergeometric distribution.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{n}}$.
4:    $\mathbf{k}$IntegerInput
On entry: the integer $k$ which defines the required probabilities.
Constraint: $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(0,{\mathbf{l}}-\left({\mathbf{n}}-{\mathbf{m}}\right)\right)\le {\mathbf{k}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{l}},{\mathbf{m}}\right)$.
5:    $\mathbf{plek}$double *Output
On exit: the lower tail probability, $\mathrm{Prob}\left\{X\le k\right\}$.
6:    $\mathbf{pgtk}$double *Output
On exit: the upper tail probability, $\mathrm{Prob}\left\{X>k\right\}$.
7:    $\mathbf{peqk}$double *Output
On exit: the point probability, $\mathrm{Prob}\left\{X=k\right\}$.
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_2_INT_ARG_GT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{l}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\le {\mathbf{l}}$.
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\le {\mathbf{m}}$.
On entry, ${\mathbf{l}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{l}}\le {\mathbf{n}}$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\le {\mathbf{n}}$.
NE_4_INT_ARG_CONS
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$, ${\mathbf{l}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{l}}+{\mathbf{m}}-{\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge {\mathbf{l}}+{\mathbf{m}}-{\mathbf{n}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARG_TOO_LARGE
On entry, n is too large to be represented exactly as a double precision number.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT_ARG_LT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 0$.
On entry, ${\mathbf{l}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{l}}\ge 0$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_VARIANCE_TOO_LARGE
On entry, the variance $\text{}=\frac{lm\left(n-l\right)\left(n-m\right)}{{n}^{2}\left(n-1\right)}$ exceeds ${10}^{6}$.

## 7Accuracy

Results are correct to a relative accuracy of at least ${10}^{-6}$ on machines with a precision of $9$ or more decimal digits, and to a relative accuracy of at least ${10}^{-3}$ on machines of lower precision (provided that the results do not underflow to zero).

## 8Parallelism and Performance

nag_hypergeom_dist (g01blc) is not threaded in any implementation.

The time taken by nag_hypergeom_dist (g01blc) depends on the variance (see Section 3) and on $k$. For given variance, the time is greatest when $k\approx lm/n$ ($=$ the mean), and is then approximately proportional to the square-root of the variance.

## 10Example

This example reads values of $n$, $l$, $m$ and $k$ from a data file until end-of-file is reached, and prints the corresponding probabilities.

### 10.1Program Text

Program Text (g01blce.c)

### 10.2Program Data

Program Data (g01blce.d)

### 10.3Program Results

Program Results (g01blce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017