NAG CL Interface
g01bjc (prob_​binomial)

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1 Purpose

g01bjc returns the lower tail, upper tail and point probabilities associated with a binomial distribution.

2 Specification

#include <nag.h>
void  g01bjc (Integer n, double p, Integer k, double *plek, double *pgtk, double *peqk, NagError *fail)
The function may be called by the names: g01bjc, nag_stat_prob_binomial or nag_binomial_dist.

3 Description

Let X denote a random variable having a binomial distribution with parameters n and p (n0 and 0<p<1). Then
Prob{X=k}=( n k ) pk(1-p)n-k,  k=0,1,,n.  
The mean of the distribution is np and the variance is np(1-p).
g01bjc computes for given n, p and k the probabilities:
plek=Prob{Xk} pgtk=Prob{X>k} peqk=Prob{X=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: n Integer Input
On entry: the parameter n of the binomial distribution.
Constraint: n0.
2: p double Input
On entry: the parameter p of the binomial distribution.
Constraint: 0.0<p<1.0.
3: k Integer Input
On entry: the integer k which defines the required probabilities.
Constraint: 0kn.
4: plek double * Output
On exit: the lower tail probability, Prob{Xk}.
5: pgtk double * Output
On exit: the upper tail probability, Prob{X>k}.
6: peqk double * Output
On exit: the point probability, Prob{X=k}.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_GT
On entry, k=value and n=value.
Constraint: kn.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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: k0.
On entry, n=value.
Constraint: n0.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_GE
On entry, p=value.
Constraint: p<1.0.
NE_REAL_ARG_LE
On entry, p=value.
Constraint: p>0.0.
NE_VARIANCE_TOO_LARGE
On entry, the variance (=np(1-p)) exceeds 106.

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

g01bjc is not threaded in any implementation.

9 Further Comments

The time taken by g01bjc depends on the variance (=np(1-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.

10 Example

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

10.1 Program Text

Program Text (g01bjce.c)

10.2 Program Data

Program Data (g01bjce.d)

10.3 Program Results

Program Results (g01bjce.r)