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_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.
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)