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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_prob_binomial (g01bj)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_prob_binomial (g01bj) returns the lower tail, upper tail and point probabilities associated with a binomial distribution.


[plek, pgtk, peqk, ifail] = g01bj(n, p, k)
[plek, pgtk, peqk, ifail] = nag_stat_prob_binomial(n, p, k)


Let X denote a random variable having a binomial distribution with parameters n and p (n0 and 0<p<1). Then
ProbX=k= n k pk1-pn-k,  k=0,1,,n.  
The mean of the distribution is np and the variance is np1-p.
nag_stat_prob_binomial (g01bj) computes for given n, p 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 binomial distribution.
Constraint: n0.
2:     p – double scalar
The parameter p of the binomial distribution.
Constraint: 0.0<p<1.0.
3:     k int64int32nag_int scalar
The integer k which defines the required probabilities.
Constraint: 0kn.

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,p0.0,
On entry,k<0,
On entry,n is too large to be represented exactly as a double number.
On entry,the variance (=np1-p) 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_binomial (g01bj) depends on the variance (=np1-p) and on k. For given variance, the time is greatest when knp (=the mean), and is then approximately proportional to the square-root of the variance.


This example reads values of n and p from a data file until end-of-file is reached, and prints the corresponding probabilities.
function g01bj_example

fprintf('g01bj example results\n\n');

n = int64([4    19     100    2000]);
k = int64([2    13     67     700]);
p =         [0.5  0.44   0.75   0.33];

fprintf('   n     p      k     plek      pgtk      peqk\n');
for i=1:4
  [plek, pgtk, peqk, ifail] = ...
  g01bj(n(i), p(i), k(i));
  fprintf('%5d%7.3f%5d%10.5f%10.5f%10.5f\n', n(i), p(i), k(i), ...
	  plek, pgtk, peqk);

g01bj example results

   n     p      k     plek      pgtk      peqk
    4  0.500    2   0.68750   0.31250   0.37500
   19  0.440   13   0.99138   0.00862   0.01939
  100  0.750   67   0.04460   0.95540   0.01700
 2000  0.330  700   0.97251   0.02749   0.00312

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

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