g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_prob_binomial_vector (g01sjc)

## 1  Purpose

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

## 2  Specification

 #include #include
 void nag_prob_binomial_vector (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)

## 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 ${n}_{i}{p}_{i}\left(1-{p}_{i}\right)$.
nag_prob_binomial_vector (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.

## 4  References

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

## 5  Arguments

1:     lnIntegerInput
On entry: the length of the array n
Constraint: ${\mathbf{ln}}>0$.
2:     n[ln]const IntegerInput
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:     lpIntegerInput
On entry: the length of the array p
Constraint: ${\mathbf{lp}}>0$.
4:     p[lp]const doubleInput
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:     lkIntegerInput
On entry: the length of the array k
Constraint: ${\mathbf{lk}}>0$.
6:     k[lk]const IntegerInput
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[$\mathit{dim}$]doubleOutput
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:     pgtk[$\mathit{dim}$]doubleOutput
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:     peqk[$\mathit{dim}$]doubleOutput
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:   ivalid[$\mathit{dim}$]IntegerOutput
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:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
NW_IVALID
On entry, at least one value of n, p or k was invalid.

## 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 nag_prob_binomial_vector (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.

## 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 (g01sjce.c)

### 9.2  Program Data

Program Data (g01sjce.d)

### 9.3  Program Results

Program Results (g01sjce.r)