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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_prob_hypergeom (g01bl)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_prob_hypergeom (g01bl) returns the lower tail, upper tail and point probabilities associated with a hypergeometric distribution.


[plek, pgtk, peqk, ifail] = g01bl(n, l, m, k)
[plek, pgtk, peqk, ifail] = nag_stat_prob_hypergeom(n, l, m, k)


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_stat_prob_hypergeom (g01bl) 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).


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


Compulsory Input Parameters

1:     n int64int32nag_int scalar
The parameter n of the hypergeometric distribution.
Constraint: n0.
2:     l int64int32nag_int scalar
The parameter l of the hypergeometric distribution.
Constraint: 0ln.
3:     m int64int32nag_int scalar
The parameter m of the hypergeometric distribution.
Constraint: 0mn.
4:     k int64int32nag_int scalar
The integer k which defines the required probabilities.
Constraint: max0,l-n-mkminl,m.

Optional Input Parameters


Output Parameters

1:     plek – double scalar
The lower tail probability, ProbXk.
2:     pgtk – double scalar
The upper tail probability, ProbX>k.
3:     peqk – double scalar
The point probability, ProbX=k.
4:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry,n<0.
On entry,l<0,
On entry,m<0,
On entry,k<0,
On entry,n is too large to be represented exactly as a double number.
On entry,the variance (see Description) exceeds 106.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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).

Further Comments

The time taken by nag_stat_prob_hypergeom (g01bl) depends on the variance (see Description) 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.


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.
function g01bl_example

fprintf('g01bl example results\n\n');

n = int64([10 40 155 1000]);
l = int64([ 2 10  35  444]);
m = int64([ 5  3 122  500]);
k = int64([ 1  2  22  220]);

fprintf('    n   l   m   k     plek      pgtk      peqk\n');
for i = 1:4
  [plek, pgtk, peqk, ifail] = ...
  g01bl(n(i), l(i), m(i), k(i));

  fprintf('%5d%4d%4d%4d%10.5f%10.5f%10.5f\n', n(i), l(i), m(i), k(i), ...
	  plek, pgtk, peqk);

g01bl example results

    n   l   m   k     plek      pgtk      peqk
   10   2   5   1   0.77778   0.22222   0.55556
   40  10   3   2   0.98785   0.01215   0.13664
  155  35 122  22   0.01101   0.98899   0.00779
 1000 444 500 220   0.42429   0.57571   0.04913

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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