# NAG FL Interfaceg01sjf (prob_​binomial_​vector)

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## 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(*)
C Header Interface
#include <nag.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)
The routine may be called by the names g01sjf or nagf_stat_prob_binomial_vector.

## 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
 $Prob{Xi=ki}=( ni ki ) piki(1-pi)ni-ki, ki=0,1,…,ni.$
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: $\mathbf{ln}$Integer Input
On entry: the length of the array n.
Constraint: ${\mathbf{ln}}>0$.
2: $\mathbf{n}\left({\mathbf{ln}}\right)$Integer array Input
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: $\mathbf{lp}$Integer Input
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4: $\mathbf{p}\left({\mathbf{lp}}\right)$Real (Kind=nag_wp) array Input
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: $\mathbf{lk}$Integer Input
On entry: the length of the array k.
Constraint: ${\mathbf{lk}}>0$.
6: $\mathbf{k}\left({\mathbf{lk}}\right)$Integer array Input
On entry: ${k}_{i}$, the integer which defines the required probabilities with ${k}_{i}={\mathbf{k}}\left(j\right)$, .
Constraint: $0\le {k}_{i}\le {n}_{i}$.
7: $\mathbf{plek}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array plek must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ln}},{\mathbf{lp}},{\mathbf{lk}}\right)$.
On exit: $\mathrm{Prob}\left\{{X}_{i}\le {k}_{i}\right\}$, the lower tail probabilities.
8: $\mathbf{pgtk}\left(*\right)$Real (Kind=nag_wp) array Output
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) array Output
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 array Output
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: $\mathbf{ifail}$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 n, p or k was invalid.
Check ivalid for more information.
${\mathbf{ifail}}=2$
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: ${\mathbf{lk}}>0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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, 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.

## 9Further Comments

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)