NAG Library Function Document

nag_hypergeom_dist (g01blc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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

(

n \geq l \geq 0

n \geq m \geq 0

). Then

Prob \{X = k\} = \frac{(\begin{array}{l} m \\ k \end{array}) (\begin{array}{l} n - m \\ l - k \end{array})}{(\begin{array}{l} n \\ l \end{array})},

where

\max (0, l - (n - m)) \leq k \leq \min (l, m)

0 \leq l \leq n

and

0 \leq m \leq 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

= \frac{l m}{n}

, and the variance

= \frac{l m (n - l) (n - m)}{n^{2} (n - 1)}

nag_hypergeom_dist (g01blc) computes for given

n

l

m

and

k

the probabilities:

\begin{matrix} plek = Prob \{X \leq k\} \\ pgtk = Prob \{X > k\} \\ peqk = Prob \{X = k\} . \end{matrix}

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: n – IntegerInput: On entry: the parameter $n$ of the hypergeometric distribution.
Constraint: $n \geq 0$ .
2: l – IntegerInput: On entry: the parameter $l$ of the hypergeometric distribution.
Constraint: $0 \leq l \leq n$ .
3: m – IntegerInput: On entry: the parameter $m$ of the hypergeometric distribution.
Constraint: $0 \leq m \leq n$ .
4: k – IntegerInput: On entry: the integer $k$ which defines the required probabilities.
Constraint: $\max (0, l - (n - m)) \leq k \leq \min (l, m)$ .
5: plek – double *Output: On exit: the lower tail probability, $Prob \{X \leq k\}$ .
6: pgtk – double *Output: On exit: the upper tail probability, $Prob \{X > k\}$ .
7: peqk – double *Output: On exit: the point probability, $Prob \{X = k\}$ .
8: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_GT: On entry, $k = ⟨value⟩$ and $l = ⟨value⟩$ .
Constraint: $k \leq l$ .
On entry, $k = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $k \leq m$ .
On entry, $l = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $l \leq n$ .
On entry, $m = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $m \leq n$ .
NE_4_INT_ARG_CONS: On entry, $k = ⟨value⟩$ , $l = ⟨value⟩$ , $m = ⟨value⟩$ and $l + m - n = ⟨value⟩$ .
Constraint: $k \geq l + m - n$ .
NE_ARG_TOO_LARGE: On entry, n is too large to be represented exactly as a double precision number.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT_ARG_LT: On entry, $k = ⟨value⟩$ .
Constraint: $k \geq 0$ .
On entry, $l = ⟨value⟩$ .
Constraint: $l \geq 0$ .
On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
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.
NE_VARIANCE_TOO_LARGE: On entry, the variance $= \frac{l m (n - l) (n - m)}{n^{2} (n - 1)}$ 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

k \approx l m / 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.

NAG Library Function Documentnag_hypergeom_dist (g01blc)