# NAG Library Routine Document

## 1Purpose

g01sjf returns a number of the lower tail, upper tail and point probabilities for the binomial distribution.

## 2Specification

Fortran Interface
 Subroutine g01sjf ( ln, n, lp, p, lk, k, plek, pgtk, peqk,
 Integer, Intent (In) :: ln, n(ln), lp, lk, k(lk) Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: p(lp) Real (Kind=nag_wp), Intent (Out) :: plek(*), pgtk(*), peqk(*)
#include <nagmk26.h>
 void g01sjf_ (const Integer *ln, const Integer n[], const Integer *lp, const double p[], const Integer *lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], Integer *ifail)

## 3Description

Let $X=\left\{{X}_{i}:i=1,2,\dots ,m\right\}$ denote a vector of random variables each having a binomial distribution with parameters ${n}_{i}$ and ${p}_{i}$ (${n}_{i}\ge 0$ and $0<{p}_{i}<1$). Then
 $ProbXi=ki= ni ki piki1-pini-ki, ki=0,1,…,ni.$
The mean of the each distribution is given by ${n}_{i}{p}_{i}$ and the variance by ${n}_{i}{p}_{i}\left(1-{p}_{i}\right)$.
g01sjf computes, for given ${n}_{i}$, ${p}_{i}$ 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 an algorithm similar to that described in Knüsel (1986) for the Poisson distribution.
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.

## 4References

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

## 5Arguments

1:     $\mathbf{ln}$ – IntegerInput
On entry: the length of the array n.
Constraint: ${\mathbf{ln}}>0$.
2:     $\mathbf{n}\left({\mathbf{ln}}\right)$ – Integer arrayInput
On entry: ${n}_{i}$, the first parameter of the binomial distribution with ${n}_{i}={\mathbf{n}}\left(j\right)$, , for $i=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ln}},{\mathbf{lp}},{\mathbf{lk}}\right)$.
Constraint: ${\mathbf{n}}\left(\mathit{j}\right)\ge 0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ln}}$.
3:     $\mathbf{lp}$ – IntegerInput
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4:     $\mathbf{p}\left({\mathbf{lp}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${p}_{i}$, the second parameter of the binomial distribution with ${p}_{i}={\mathbf{p}}\left(j\right)$, .
Constraint: $0.0<{\mathbf{p}}\left(\mathit{j}\right)<1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lp}}$.
5:     $\mathbf{lk}$ – IntegerInput
On entry: the length of the array k.
Constraint: ${\mathbf{lk}}>0$.
6:     $\mathbf{k}\left({\mathbf{lk}}\right)$ – Integer arrayInput
On entry: ${k}_{i}$, the integer which defines the required probabilities with ${k}_{i}={\mathbf{k}}\left(j\right)$, .
Constraint: $0\le {k}_{i}\le {n}_{i}$.
7:     $\mathbf{plek}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array plek must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ln}},{\mathbf{lp}},{\mathbf{lk}}\right)$.
On exit: $\mathrm{Prob}\left\{{X}_{i}\le {k}_{i}\right\}$, the lower tail probabilities.
8:     $\mathbf{pgtk}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array pgtk must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ln}},{\mathbf{lp}},{\mathbf{lk}}\right)$.
On exit: $\mathrm{Prob}\left\{{X}_{i}>{k}_{i}\right\}$, the upper tail probabilities.
9:     $\mathbf{peqk}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array peqk must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ln}},{\mathbf{lp}},{\mathbf{lk}}\right)$.
On exit: $\mathrm{Prob}\left\{{X}_{i}={k}_{i}\right\}$, the point probabilities.
10:   $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ln}},{\mathbf{lp}},{\mathbf{lk}}\right)$.
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, ${n}_{i}<0$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, ${p}_{i}\le 0.0$, or ${p}_{i}\ge 1.0$.
${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ${k}_{i}<0$, or ${k}_{i}>{n}_{i}$.
${\mathbf{ivalid}}\left(i\right)=4$
 On entry, ${n}_{i}$ is too large to be represented exactly as a real number.
${\mathbf{ivalid}}\left(i\right)=5$
 On entry, the variance ($\text{}={n}_{i}{p}_{i}\left(1-{p}_{i}\right)$) exceeds ${10}^{6}$.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  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 n, p or k was invalid.
${\mathbf{ifail}}=2$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ln}}>0$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lp}}>0$.
${\mathbf{ifail}}=4$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lk}}>0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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, and to a relative accuracy of at least ${10}^{-3}$ on machines of lower precision (provided that the results do not underflow to zero).

## 8Parallelism and Performance

g01sjf is not threaded in any implementation.

The time taken by g01sjf to calculate each probability depends on the variance ($\text{}={n}_{i}{p}_{i}\left(1-{p}_{i}\right)$) and on ${k}_{i}$. For given variance, the time is greatest when ${k}_{i}\approx {n}_{i}{p}_{i}$ ($\text{}=\text{the mean}$), and is then approximately proportional to the square-root of the variance.

## 10Example

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

### 10.1Program Text

Program Text (g01sjfe.f90)

### 10.2Program Data

Program Data (g01sjfe.d)

### 10.3Program Results

Program Results (g01sjfe.r)