# NAG FL Interfaceg01bjf (prob_​binomial)

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## 1Purpose

g01bjf returns the lower tail, upper tail and point probabilities associated with a binomial distribution.

## 2Specification

Fortran Interface
 Subroutine g01bjf ( n, p, k, plek, pgtk, peqk,
 Integer, Intent (In) :: n, k Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: p Real (Kind=nag_wp), Intent (Out) :: plek, pgtk, peqk
#include <nag.h>
 void g01bjf_ (const Integer *n, const double *p, const Integer *k, double *plek, double *pgtk, double *peqk, Integer *ifail)
The routine may be called by the names g01bjf or nagf_stat_prob_binomial.

## 3Description

Let $X$ denote a random variable having a binomial distribution with parameters $n$ and $p$ ($n\ge 0$ and $0). Then
 $Prob{X=k}=( n k ) pk(1-p)n-k, k=0,1,…,n.$
The mean of the distribution is $np$ and the variance is $np\left(1-p\right)$.
g01bjf computes for given $n$, $p$ and $k$ the probabilities:
 $plek=Prob{X≤k} pgtk=Prob{X>k} peqk=Prob{X=k} .$
The method is similar to the method for the Poisson distribution described in Knüsel (1986).

## 4References

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

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: the parameter $n$ of the binomial distribution.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{p}$Real (Kind=nag_wp) Input
On entry: the parameter $p$ of the binomial distribution.
Constraint: $0.0<{\mathbf{p}}<1.0$.
3: $\mathbf{k}$Integer Input
On entry: the integer $k$ which defines the required probabilities.
Constraint: $0\le {\mathbf{k}}\le {\mathbf{n}}$.
4: $\mathbf{plek}$Real (Kind=nag_wp) Output
On exit: the lower tail probability, $\mathrm{Prob}\left\{X\le k\right\}$.
5: $\mathbf{pgtk}$Real (Kind=nag_wp) Output
On exit: the upper tail probability, $\mathrm{Prob}\left\{X>k\right\}$.
6: $\mathbf{peqk}$Real (Kind=nag_wp) Output
On exit: the point probability, $\mathrm{Prob}\left\{X=k\right\}$.
7: $\mathbf{ifail}$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, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}<1.0$.
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}>0.0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{k}}\ge 0$.
On entry, ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{k}}\le {\mathbf{n}}$.
${\mathbf{ifail}}=4$
On entry, n is too large to be represented exactly as a double precision number.
${\mathbf{ifail}}=5$
On entry, the variance $\left(=np\left(1-p\right)\right)$ exceeds ${10}^{6}$.
${\mathbf{ifail}}=-99$
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

g01bjf is not threaded in any implementation.

The time taken by g01bjf 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.

## 10Example

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

### 10.1Program Text

Program Text (g01bjfe.f90)

### 10.2Program Data

Program Data (g01bjfe.d)

### 10.3Program Results

Program Results (g01bjfe.r)