# NAG FL Interfaceg01skf (prob_​poisson_​vector)

## 1Purpose

g01skf returns a number of the lower tail, upper tail and point probabilities for the Poisson distribution.

## 2Specification

Fortran Interface
 Subroutine g01skf ( ll, l, lk, k, plek, pgtk, peqk,
 Integer, Intent (In) :: ll, lk, k(lk) Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: l(ll) Real (Kind=nag_wp), Intent (Out) :: plek(*), pgtk(*), peqk(*)
#include <nag.h>
 void g01skf_ (const Integer *ll, const double l[], const Integer *lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], Integer *ifail)
The routine may be called by the names g01skf or nagf_stat_prob_poisson_vector.

## 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 ${\lambda }_{i}$.
g01skf 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 routine 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.
Knüsel L (1986) Computation of the chi-square and Poisson distribution SIAM J. Sci. Statist. Comput. 7 1022–1036

## 5Arguments

1: $\mathbf{ll}$Integer Input
On entry: the length of the array l.
Constraint: ${\mathbf{ll}}>0$.
2: $\mathbf{l}\left({\mathbf{ll}}\right)$Real (Kind=nag_wp) array Input
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: $0.0<{\mathbf{l}}\left(\mathit{j}\right)\le {10}^{6}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ll}}$.
3: $\mathbf{lk}$Integer Input
On entry: the length of the array k.
Constraint: ${\mathbf{lk}}>0$.
4: $\mathbf{k}\left({\mathbf{lk}}\right)$Integer array Input
On entry: ${k}_{i}$, the integer which defines the required probabilities with ${k}_{i}={\mathbf{k}}\left(j\right)$, .
Constraint: ${\mathbf{k}}\left(\mathit{j}\right)\ge 0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lk}}$.
5: $\mathbf{plek}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension 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: $\mathbf{pgtk}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension 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: $\mathbf{peqk}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension 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: $\mathbf{ivalid}\left(*\right)$Integer array Output
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ll}},{\mathbf{lk}}\right)$.
On exit: ${\mathbf{ivalid}}\left(i\right)$ 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, ${\lambda }_{i}>{10}^{6}$.
9: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one value of l or k was invalid.
${\mathbf{ifail}}=2$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ll}}>0$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lk}}>0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 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

g01skf is not threaded in any implementation.

The time taken by g01skf 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 (g01skfe.f90)

### 10.2Program Data

Program Data (g01skfe.d)

### 10.3Program Results

Program Results (g01skfe.r)