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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_prob_poisson_vector (g01sk)

## Purpose

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

## Syntax

[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)

## Description

Let $X=\left\{{X}_{i}:i=1,2,\dots ,m\right\}$ denote a vector of random variables each having a Poisson distribution with parameter ${\lambda }_{i}$ $\left(>0\right)$. Then
 $Prob Xi = ki = e -λi λi ki ki! , ki = 0,1,2,…$
The mean and variance of each distribution are both equal to ${\lambda }_{i}$.
nag_stat_prob_poisson_vector (g01sk) computes, for given ${\lambda }_{i}$ and ${k}_{i}$ the probabilities: $\mathrm{Prob}\left\{{X}_{i}\le {k}_{i}\right\}$, $\mathrm{Prob}\left\{{X}_{i}>{k}_{i}\right\}$ and $\mathrm{Prob}\left\{{X}_{i}={k}_{i}\right\}$ 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.

## 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:     $\mathrm{l}\left({\mathbf{ll}}\right)$ – double array
${\lambda }_{i}$, the parameter of the Poisson distribution with ${\lambda }_{i}={\mathbf{l}}\left(j\right)$, , for $i=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$.
Constraint: $0.0<{\mathbf{l}}\left(\mathit{j}\right)\le {10}^{6}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ll}}$.
2:     $\mathrm{k}\left({\mathbf{lk}}\right)$int64int32nag_int array
${k}_{i}$, the integer which defines the required probabilities with ${k}_{i}={\mathbf{k}}\left(j\right)$, .
Constraint: ${\mathbf{k}}\left(\mathit{j}\right)\ge 0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lk}}$.

### Optional Input Parameters

1:     $\mathrm{ll}$int64int32nag_int scalar
Default: the dimension of the array l.
The length of the array l
Constraint: ${\mathbf{ll}}>0$.
2:     $\mathrm{lk}$int64int32nag_int scalar
Default: the dimension of the array k.
The length of the array k
Constraint: ${\mathbf{lk}}>0$.

### Output Parameters

1:     $\mathrm{plek}\left(:\right)$ – double array
The dimension of the array plek will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$
$\mathrm{Prob}\left\{{X}_{i}\le {k}_{i}\right\}$, the lower tail probabilities.
2:     $\mathrm{pgtk}\left(:\right)$ – double array
The dimension of the array pgtk will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$
$\mathrm{Prob}\left\{{X}_{i}>{k}_{i}\right\}$, the upper tail probabilities.
3:     $\mathrm{peqk}\left(:\right)$ – double array
The dimension of the array peqk will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$
$\mathrm{Prob}\left\{{X}_{i}={k}_{i}\right\}$, the point probabilities.
4:     $\mathrm{ivalid}\left(:\right)$int64int32nag_int array
The dimension of the array ivalid will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$
${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
 On entry, ${\lambda }_{i}\le 0.0$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, ${k}_{i}<0$.
${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ${\lambda }_{i}>{10}^{6}$.
5:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ 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  ${\mathbf{ifail}}=1$
On entry, at least one value of l or k was invalid.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{ll}}>0$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{lk}}>0$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
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 (provided that the results do not underflow to zero).

The time taken by nag_stat_prob_poisson_vector (g01sk) to calculate each probability depends on ${\lambda }_{i}$ and ${k}_{i}$. For given ${\lambda }_{i}$, the time is greatest when ${k}_{i}\approx {\lambda }_{i}$, and is then approximately proportional to $\sqrt{{\lambda }_{i}}$.

## Example

This example reads a vector of values for $\lambda$ 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));
end

```
```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
```