# NAG C Library Function Document

## 1Purpose

nag_prob_poisson_vector (g01skc) returns a number of the lower tail, upper tail and point probabilities for the Poisson distribution.

## 2Specification

 #include #include
 void nag_prob_poisson_vector (Integer ll, const double l[], Integer lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], NagError *fail)

## 3Description

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_prob_poisson_vector (g01skc) 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 Section 2.6 in the g01 Chapter Introduction for further information.

## 4References

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

## 5Arguments

1:    $\mathbf{ll}$IntegerInput
On entry: the length of the array l.
Constraint: ${\mathbf{ll}}>0$.
2:    $\mathbf{l}\left[{\mathbf{ll}}\right]$const doubleInput
On entry: ${\lambda }_{i}$, 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: $0.0<{\mathbf{l}}\left[\mathit{j}-1\right]\le {10}^{6}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ll}}$.
3:    $\mathbf{lk}$IntegerInput
On entry: the length of the array k.
Constraint: ${\mathbf{lk}}>0$.
4:    $\mathbf{k}\left[{\mathbf{lk}}\right]$const IntegerInput
On entry: ${k}_{i}$, the integer which defines the required probabilities with ${k}_{i}={\mathbf{k}}\left[j\right]$, .
Constraint: ${\mathbf{k}}\left[\mathit{j}-1\right]\ge 0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lk}}$.
5:    $\mathbf{plek}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array plek must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$.
On exit: $\mathrm{Prob}\left\{{X}_{i}\le {k}_{i}\right\}$, the lower tail probabilities.
6:    $\mathbf{pgtk}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array pgtk must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$.
On exit: $\mathrm{Prob}\left\{{X}_{i}>{k}_{i}\right\}$, the upper tail probabilities.
7:    $\mathbf{peqk}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array peqk must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$.
On exit: $\mathrm{Prob}\left\{{X}_{i}={k}_{i}\right\}$, the point probabilities.
8:    $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, ${\lambda }_{i}\le 0.0$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, ${k}_{i}<0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
 On entry, ${\lambda }_{i}>{10}^{6}$.
9:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lk}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ll}}>0$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID
On entry, at least one value of l or k was invalid.

## 7Accuracy

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).

## 8Parallelism and Performance

nag_prob_poisson_vector (g01skc) is not threaded in any implementation.

The time taken by nag_prob_poisson_vector (g01skc) 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 $\sqrt{{\lambda }_{i}}$.

## 10Example

This example reads a vector of values for $\lambda$ and $k$, and prints the corresponding probabilities.

### 10.1Program Text

Program Text (g01skce.c)

### 10.2Program Data

Program Data (g01skce.d)

### 10.3Program Results

Program Results (g01skce.r)