nag_hypergeom_dist (g01blc) (PDF version)
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g01 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_hypergeom_dist (g01blc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagg01.h>
void  nag_hypergeom_dist (Integer n, Integer l, Integer m, Integer k, double *plek, double *pgtk, double *peqk, NagError *fail)

3  Description

Let X denote a random variable having a hypergeometric distribution with parameters n, l and m (nl0, nm0). Then
ProbX=k= m k n-m l-k n l ,
where max0,l-n-m k minl,m , 0ln and 0mn.
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 = lm n , and the variance = lmn-ln-m n2n-1 .
nag_hypergeom_dist (g01blc) computes for given n, l, m and k the probabilities:
plek=ProbXk pgtk=ProbX>k peqk=ProbX=k .
The method is similar to the method for the Poisson distribution described in Knüsel (1986).

4  References

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

5  Arguments

1:     nIntegerInput
On entry: the parameter n of the hypergeometric distribution.
Constraint: n0.
2:     lIntegerInput
On entry: the parameter l of the hypergeometric distribution.
Constraint: 0ln.
3:     mIntegerInput
On entry: the parameter m of the hypergeometric distribution.
Constraint: 0mn.
4:     kIntegerInput
On entry: the integer k which defines the required probabilities.
Constraint: max0,l-n-mkminl,m.
5:     plekdouble *Output
On exit: the lower tail probability, ProbXk.
6:     pgtkdouble *Output
On exit: the upper tail probability, ProbX>k.
7:     peqkdouble *Output
On exit: the point probability, ProbX=k.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, k=value and l=value.
Constraint: kl.
On entry, k=value and m=value.
Constraint: km.
On entry, l=value and n=value.
Constraint: ln.
On entry, m=value and n=value.
Constraint: mn.
On entry, k=value, l=value, m=value and l+m-n=value.
Constraint: kl+m-n.
On entry, n is too large to be represented exactly as a double precision number.
On entry, argument value had an illegal value.
On entry, k=value.
Constraint: k0.
On entry, l=value.
Constraint: l0.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
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.
On entry, the variance = lm n-l n-m n 2 n-1  exceeds 10 6 .

7  Accuracy

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).

8  Parallelism and Performance

Not applicable.

9  Further Comments

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 klm/n (= the mean), and is then approximately proportional to the square-root of the variance.

10  Example

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.

10.1  Program Text

Program Text (g01blce.c)

10.2  Program Data

Program Data (g01blce.d)

10.3  Program Results

Program Results (g01blce.r)

nag_hypergeom_dist (g01blc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014