NAG Library Function Document

nag_poisson_dist (g01bkc)

▸▿ 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_poisson_dist (g01bkc) returns the lower tail, upper tail and point probabilities associated with a Poisson distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

void	nag_poisson_dist (double rlamda, Integer k, double plek, double pgtk, double peqk, NagError fail)

3 Description

Let

X

denote a random variable having a Poisson distribution with parameter

λ

(> 0)

. Then

Prob \{X = k\} = e^{- λ} \frac{λ^{k}}{k!}, k = 0, 1, 2, \dots

The mean and variance of the distribution are both equal to

λ

nag_poisson_dist (g01bkc) computes for given

λ

and

k

the probabilities:

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

The method is 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: $rlamda$ – doubleInput: On entry: the parameter $λ$ of the Poisson distribution.

Constraint: $0.0 < rlamda \leq 10^{6}$ .
2: $k$ – IntegerInput: On entry: the integer $k$ which defines the required probabilities.

Constraint: $k \geq 0$ .
3: $plek$ – double *Output: On exit: the lower tail probability, $Prob \{X \leq k\}$ .
4: $pgtk$ – double *Output: On exit: the upper tail probability, $Prob \{X > k\}$ .
5: $peqk$ – double *Output: On exit: the point probability, $Prob \{X = k\}$ .
6: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT_ARG_LT: On entry, $k = 〈value〉$ .
Constraint: $k \geq 0$ .
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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL_ARG_GT: On entry, $rlamda = 〈value〉$ .
Constraint: $rlamda \leq 10^{6}$ .
NE_REAL_ARG_LE: On entry, $rlamda = 〈value〉$ .
Constraint: $rlamda > 0.0$ .

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_poisson_dist (g01bkc) depends on

λ

and

k

. For given

λ

, the time is greatest when

k \approx λ

, and is then approximately proportional to

\sqrt{λ}

10 Example

This example reads values of

λ

and

k

from a data file until end-of-file is reached, and prints the corresponding probabilities.

NAG Library Function Documentnag_poisson_dist (g01bkc)