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NAG Toolbox

NAG Toolbox: nag_stat_prob_poisson (g01bk)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_stat_prob_poisson (g01bk) returns the lower tail, upper tail and point probabilities associated with a Poisson distribution.

Syntax

[plek, pgtk, peqk, ifail] = g01bk(rlamda, k)
[plek, pgtk, peqk, ifail] = nag_stat_prob_poisson(rlamda, k)

Description

Let X denote a random variable having a Poisson distribution with parameter λ >0. Then
ProbX=k=e-λλkk! ,  k=0,1,2,  
The mean and variance of the distribution are both equal to λ.
nag_stat_prob_poisson (g01bk) computes for given λ and k the probabilities:
plek=ProbXk pgtk=ProbX>k peqk=ProbX=k .  
The method is described in Knüsel (1986).

References

Knüsel L (1986) Computation of the chi-square and Poisson distribution SIAM J. Sci. Statist. Comput. 7 1022–1036

Parameters

Compulsory Input Parameters

1:     rlamda – double scalar
The parameter λ of the Poisson distribution.
Constraint: 0.0<rlamda106.
2:     k int64int32nag_int scalar
The integer k which defines the required probabilities.
Constraint: k0.

Optional Input Parameters

None.

Output Parameters

1:     plek – double scalar
The lower tail probability, ProbXk.
2:     pgtk – double scalar
The upper tail probability, ProbX>k.
3:     peqk – double scalar
The point probability, ProbX=k.
4:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry,rlamda0.0.
   ifail=2
On entry,k<0.
   ifail=3
On entry,rlamda>106.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

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).

Further Comments

The time taken by nag_stat_prob_poisson (g01bk) depends on λ and k. For given λ, the time is greatest when kλ, and is then approximately proportional to λ.

Example

This example reads values of λ and k from a data file until end-of-file is reached, and prints the corresponding probabilities.
function g01bk_example


fprintf('g01bk example results\n\n');

rlamda =    [0.75  9.2  34  175];
k = int64([3     12   25  175]);

fprintf('  rlamda     k     plek      pgtk      peqk\n');
for i=1:4
  [plek, pgtk, peqk, ifail] = ...
  g01bk(rlamda(i), k(i));

  fprintf('%8.3f%6d%10.5f%10.5f%10.5f\n', rlamda(i), k(i), plek, pgtk, peqk);
end


g01bk example results

  rlamda     k     plek      pgtk      peqk
   0.750     3   0.99271   0.00729   0.03321
   9.200    12   0.86074   0.13926   0.07755
  34.000    25   0.06736   0.93264   0.02140
 175.000   175   0.52009   0.47991   0.03014

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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