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NAG Library Function Document

nag_binomial_dist (g01bjc)

▸▿ 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_binomial_dist (g01bjc) returns the lower tail, upper tail and point probabilities associated with a binomial distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

void	nag_binomial_dist (Integer n, double p, Integer k, double plek, double pgtk, double peqk, NagError fail)

3 Description

Let

X

denote a random variable having a binomial distribution with parameters

n

and

p

(

n \geq 0

and

0 < p < 1

). Then

Prob \{X = k\} = (\begin{array}{l} n \\ k \end{array}) p^{k} {(1 - p)}^{n - k}, k = 0, 1, \dots, n .

The mean of the distribution is

n p

and the variance is

n p (1 - p)

nag_binomial_dist (g01bjc) computes for given

n

p

and

k

the probabilities:

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

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 binomial distribution.

Constraint: $n \geq 0$ .
2: $p$ – doubleInput: On entry: the parameter $p$ of the binomial distribution.

Constraint: $0.0 < p < 1.0$ .
3: $k$ – IntegerInput: On entry: the integer $k$ which defines the required probabilities.

Constraint: $0 \leq k \leq n$ .
4: $plek$ – double *Output: On exit: the lower tail probability, $Prob \{X \leq k\}$ .
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 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_2_INT_ARG_GT: On entry, $k = 〈value〉$ and $n = 〈value〉$ .
Constraint: $k \leq$ or $n$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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: $k \geq 0$ .

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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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 $(= n p (1 - p))$ 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

nag_binomial_dist (g01bjc) is not threaded in any implementation.

9 Further Comments

The time taken by nag_binomial_dist (g01bjc) depends on the variance (

= n p (1 - p)

) and on

k

. For given variance, the time is greatest when

k \approx n p

(

= 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.

NAG Library Function Documentnag_binomial_dist (g01bjc)