g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_binomial_dist (g01bjc)

## 1  Purpose

nag_binomial_dist (g01bjc) returns the lower tail, upper tail and point probabilities associated with a binomial distribution.

## 2  Specification

 #include #include
 void nag_binomial_dist (Integer n, double p, Integer k, double *plek, double *pgtk, double *peqk, NagError *fail)

## 3  Description

Let $X$ denote a random variable having a binomial distribution with parameters $n$ and $p$ ($n\ge 0$ and $0). Then
 $ProbX=k= n k pk1-pn-k, k=0,1,…,n.$
The mean of the distribution is $np$ and the variance is $np\left(1-p\right)$.
nag_binomial_dist (g01bjc) computes for given $n$, $p$ and $k$ the probabilities:
 $plek=ProbX≤k pgtk=ProbX>k peqk=ProbX=k .$
The method is similar to the method for the Poisson distribution described in Knüsel (1986).
Knüsel L (1986) Computation of the chi-square and Poisson distribution SIAM J. Sci. Statist. Comput. 7 1022–1036

## 5  Arguments

1:    $\mathbf{n}$IntegerInput
On entry: the parameter $n$ of the binomial distribution.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{p}$doubleInput
On entry: the parameter $p$ of the binomial distribution.
Constraint: $0.0<{\mathbf{p}}<1.0$.
3:    $\mathbf{k}$IntegerInput
On entry: the integer $k$ which defines the required probabilities.
Constraint: $0\le {\mathbf{k}}\le {\mathbf{n}}$.
4:    $\mathbf{plek}$double *Output
On exit: the lower tail probability, $\mathrm{Prob}\left\{X\le k\right\}$.
5:    $\mathbf{pgtk}$double *Output
On exit: the upper tail probability, $\mathrm{Prob}\left\{X>k\right\}$.
6:    $\mathbf{peqk}$double *Output
On exit: the point probability, $\mathrm{Prob}\left\{X=k\right\}$.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_2_INT_ARG_GT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\le$ or ${\mathbf{n}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_ARG_TOO_LARGE
On entry, n is too large to be represented exactly as a double precision number.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT_ARG_LT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL_ARG_GE
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}<1.0$.
NE_REAL_ARG_LE
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}>0.0$.
NE_VARIANCE_TOO_LARGE
On entry, the variance $\left(=np\left(1-p\right)\right)$ exceeds ${10}^{6}$.

## 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).

## 8  Parallelism and Performance

Not applicable.

The time taken by nag_binomial_dist (g01bjc) depends on the variance ($\text{}=np\left(1-p\right)$) and on $k$. For given variance, the time is greatest when $k\approx np$ ($\text{}=\text{the mean}$), and is then approximately proportional to the square-root of the variance.

## 10  Example

This example reads values of $n$ and $p$ from a data file until end-of-file is reached, and prints the corresponding probabilities.

### 10.1  Program Text

Program Text (g01bjce.c)

### 10.2  Program Data

Program Data (g01bjce.d)

### 10.3  Program Results

Program Results (g01bjce.r)