g01 Chapter Contents
g01 Chapter Introduction
NAG C 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).

## 4  References

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

## 5  Arguments

1:     nIntegerInput
On entry: the parameter $n$ of the binomial distribution.
Constraint: ${\mathbf{n}}\ge 0$.
2:     pdoubleInput
On entry: the parameter $p$ of the binomial distribution.
Constraint: $0.0<{\mathbf{p}}<1.0$.
3:     kIntegerInput
On entry: the integer $k$ which defines the required probabilities.
Constraint: $0\le {\mathbf{k}}\le {\mathbf{n}}$.
4:     plekdouble *Output
On exit: the lower tail probability, $\mathrm{Prob}\left\{X\le k\right\}$.
5:     pgtkdouble *Output
On exit: the upper tail probability, $\mathrm{Prob}\left\{X>k\right\}$.
6:     peqkdouble *Output
On exit: the point probability, $\mathrm{Prob}\left\{X=k\right\}$.
7:     failNagError *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_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.
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).

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.

## 9  Example

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

### 9.1  Program Text

Program Text (g01bjce.c)

### 9.2  Program Data

Program Data (g01bjce.d)

### 9.3  Program Results

Program Results (g01bjce.r)