# NAG CL Interfaceg01sjc (prob_​binomial_​vector)

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## 1Purpose

g01sjc returns a number of the lower tail, upper tail and point probabilities for the binomial distribution.

## 2Specification

 #include
 void g01sjc (Integer ln, const Integer n[], Integer lp, const double p[], Integer lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], NagError *fail)
The function may be called by the names: g01sjc, nag_stat_prob_binomial_vector or nag_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 ${n}_{i}{p}_{i}\left(1-{p}_{i}\right)$.
g01sjc 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 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.
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: $\mathbf{n}\left[{\mathbf{ln}}\right]$const Integer 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: ${\mathbf{n}}\left[\mathit{j}-1\right]\ge 0$, 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: $\mathbf{p}\left[{\mathbf{lp}}\right]$const double Input
On entry: ${p}_{i}$, the second parameter of the binomial distribution with ${p}_{i}={\mathbf{p}}\left[j\right]$, .
Constraint: $0.0<{\mathbf{p}}\left[\mathit{j}-1\right]<1.0$, 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: $\mathbf{k}\left[{\mathbf{lk}}\right]$const Integer 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: $\mathbf{plek}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, 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: $\mathbf{pgtk}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, 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: $\mathbf{peqk}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, 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: $\mathbf{ivalid}\left[\mathit{dim}\right]$Integer Output
Note: the dimension, dim, 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: ${\mathbf{ivalid}}\left[i-1\right]$ 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, ${n}_{i}<0$.
${\mathbf{ivalid}}\left[i-1\right]=2$
On entry, ${p}_{i}\le 0.0$, or, ${p}_{i}\ge 1.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
On entry, ${k}_{i}<0$, or, ${k}_{i}>{n}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=4$
On entry, ${n}_{i}$ is too large to be represented exactly as a real number.
${\mathbf{ivalid}}\left[i-1\right]=5$
On entry, the variance ($\text{}={n}_{i}{p}_{i}\left(1-{p}_{i}\right)$) exceeds ${10}^{6}$.
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lk}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ln}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lp}}>0$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of n, p or k was invalid.

## 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

g01sjc is not threaded in any implementation.

The time taken by g01sjc 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 (g01sjce.c)

### 10.2Program Data

Program Data (g01sjce.d)

### 10.3Program Results

Program Results (g01sjce.r)