G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01SJF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01SJF returns a number of the lower tail, upper tail and point probabilities for the binomial distribution.

## 2  Specification

 SUBROUTINE G01SJF ( LN, N, LP, P, LK, K, PLEK, PGTK, PEQK, IVALID, IFAIL)
 INTEGER LN, N(LN), LP, LK, K(LK), IVALID(*), IFAIL REAL (KIND=nag_wp) P(LP), PLEK(*), PGTK(*), PEQK(*)

## 3  Description

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 parameters 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  Parameters

1:     LN – INTEGERInput
On entry: the length of the array N
Constraint: ${\mathbf{LN}}>0$.
2:     N(LN) – 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:     LP – INTEGERInput
On entry: the length of the array P
Constraint: ${\mathbf{LP}}>0$.
4:     P(LP) – 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:     LK – INTEGERInput
On entry: the length of the array K
Constraint: ${\mathbf{LK}}>0$.
6:     K(LK) – 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:     PLEK($*$) – 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:     PGTK($*$) – 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:     PEQK($*$) – 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:   IVALID($*$) – 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:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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: ${\mathbf{LK}}>0$.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.

## 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, and to a relative accuracy of at least ${10}^{-3}$ on machines of lower precision (provided that the results do not underflow to zero).

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.

## 9  Example

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

### 9.1  Program Text

Program Text (g01sjfe.f90)

### 9.2  Program Data

Program Data (g01sjfe.d)

### 9.3  Program Results

Program Results (g01sjfe.r)