# NAG Library Routine Document

## 1Purpose

g01blf returns the lower tail, upper tail and point probabilities associated with a hypergeometric distribution.

## 2Specification

Fortran Interface
 Subroutine g01blf ( n, l, m, k, plek, pgtk, peqk,
 Integer, Intent (In) :: n, l, m, k Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Out) :: plek, pgtk, peqk
#include <nagmk26.h>
 void g01blf_ (const Integer *n, const Integer *l, const Integer *m, const Integer *k, double *plek, double *pgtk, double *peqk, Integer *ifail)

## 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 $\text{}=\frac{lm\left(n-l\right)\left(n-m\right)}{{n}^{2}\left(n-1\right)}$.
g01blf 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:     $\mathbf{plek}$ – Real (Kind=nag_wp)Output
On exit: the lower tail probability, $\mathrm{Prob}\left\{X\le k\right\}$.
6:     $\mathbf{pgtk}$ – Real (Kind=nag_wp)Output
On exit: the upper tail probability, $\mathrm{Prob}\left\{X>k\right\}$.
7:     $\mathbf{peqk}$ – Real (Kind=nag_wp)Output
On exit: the point probability, $\mathrm{Prob}\left\{X=k\right\}$.
8:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{l}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{l}}\ge 0$.
On entry, ${\mathbf{l}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{l}}\le {\mathbf{n}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\le {\mathbf{n}}$.
${\mathbf{ifail}}=4$
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: ${\mathbf{k}}\ge {\mathbf{l}}+{\mathbf{m}}-{\mathbf{n}}$.
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 0$.
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: ${\mathbf{k}}\le {\mathbf{m}}$.
${\mathbf{ifail}}=5$
On entry, n is too large to be represented exactly as a double precision number.
${\mathbf{ifail}}=6$
On entry, the variance $\text{}=\frac{lm\left(n-l\right)\left(n-m\right)}{{n}^{2}\left(n-1\right)}$ exceeds ${10}^{6}$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 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

g01blf is not threaded in any implementation.

The time taken by g01blf 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 (g01blfe.f90)

### 10.2Program Data

Program Data (g01blfe.d)

### 10.3Program Results

Program Results (g01blfe.r)