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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_GT: On entry, $k = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $k \leq$ or $n$ .
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, $n = ⟨value⟩$ .
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_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

Not applicable.

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)