g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_prob_poisson_vector (g01skc)

## 1  Purpose

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

## 2  Specification

 #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)

## 3  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_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.

## 4  References

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

## 5  Arguments

1:     llIntegerInput
On entry: the length of the array l
Constraint: ${\mathbf{ll}}>0$.
2:     l[ll]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:     lkIntegerInput
On entry: the length of the array k
Constraint: ${\mathbf{lk}}>0$.
4:     k[lk]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:     plek[$\mathit{dim}$]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:     pgtk[$\mathit{dim}$]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:     peqk[$\mathit{dim}$]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:     ivalid[$\mathit{dim}$]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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
NW_IVALID
On entry, at least one value of l or k was invalid.

## 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 (provided that the results do not underflow to zero).

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 $\sqrt{{\lambda }_{i}}$.

## 9  Example

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

### 9.1  Program Text

Program Text (g01skce.c)

### 9.2  Program Data

Program Data (g01skce.d)

### 9.3  Program Results

Program Results (g01skce.r)