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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_prob_poisson_vector (g01sk)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_prob_poisson_vector (g01sk) returns a number of the lower tail, upper tail and point probabilities for the Poisson distribution.


[plek, pgtk, peqk, ivalid, ifail] = g01sk(l, k, 'll', ll, 'lk', lk)
[plek, pgtk, peqk, ivalid, ifail] = nag_stat_prob_poisson_vector(l, k, 'll', ll, 'lk', lk)


Let X = Xi: i=1 , 2 ,, m  denote a vector of random variables each having a Poisson distribution with parameter λi >0. Then
Prob Xi = ki = e -λi λi ki ki! ,   ki = 0,1,2,  
The mean and variance of each distribution are both equal to λi.
nag_stat_prob_poisson_vector (g01sk) computes, for given λi and ki the probabilities: ProbXiki, ProbXi>ki and ProbXi=ki using the algorithm described in Knüsel (1986).
The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Vectorized Routines in the G01 Chapter Introduction for further information.


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


Compulsory Input Parameters

1:     lll – double array
λi, the parameter of the Poisson distribution with λi=lj, j=i-1 mod ll+1, for i=1,2,,maxll,lk.
Constraint: 0.0<lj106, for j=1,2,,ll.
2:     klk int64int32nag_int array
ki, the integer which defines the required probabilities with ki=kj, j=i-1 mod lk+1.
Constraint: kj0, for j=1,2,,lk.

Optional Input Parameters

1:     ll int64int32nag_int scalar
Default: the dimension of the array l.
The length of the array l
Constraint: ll>0.
2:     lk int64int32nag_int scalar
Default: the dimension of the array k.
The length of the array k
Constraint: lk>0.

Output Parameters

1:     plek: – double array
The dimension of the array plek will be maxll,lk
Prob Xi ki , the lower tail probabilities.
2:     pgtk: – double array
The dimension of the array pgtk will be maxll,lk
Prob Xi > ki , the upper tail probabilities.
3:     peqk: – double array
The dimension of the array peqk will be maxll,lk
Prob Xi = ki , the point probabilities.
4:     ivalid: int64int32nag_int array
The dimension of the array ivalid will be maxll,lk
ivalidi indicates any errors with the input arguments, with
No error.
On entry,λi0.0.
On entry,ki<0.
On entry,λi>106.
5:     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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ifail=1
On entry, at least one value of l or k was invalid.
Check ivalid for more information.
Constraint: ll>0.
Constraint: lk>0.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Results are correct to a relative accuracy of at least 10-6 on machines with a precision of 9 or more decimal digits (provided that the results do not underflow to zero).

Further Comments

The time taken by nag_stat_prob_poisson_vector (g01sk) to calculate each probability depends on λi and ki. For given λi, the time is greatest when kiλi, and is then approximately proportional to λi.


This example reads a vector of values for λ and k, and prints the corresponding probabilities.
function g01sk_example

fprintf('g01sk example results\n\n');

rlamda = [     0.750;  9.200; 34.000; 175.000];
k      = [int64(3); 12;     25;     175];

[plek, pgtk, peqk, ivalid, ifail] = ...
  g01sk(rlamda, k);

fprintf('    rlamda     k     plek      pgtk      peqk\n');
lrlamda = numel(rlamda);
lk      = numel(k);
len     = max ([lrlamda, lk]);
for i=0:len-1
  fprintf('%10.3f%6d%10.5f%10.5f%10.5f\n', rlamda(mod(i,lrlamda)+1), ...
          k(mod(i,lk)+1), plek(i+1), pgtk(i+1), peqk(i+1));

g01sk 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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