g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_prob_beta_dist (g01eec)

## 1  Purpose

nag_prob_beta_dist (g01eec) computes the upper and lower tail probabilities and the probability density function of the beta distribution with parameters $a$ and $b$.

## 2  Specification

 #include #include
 void nag_prob_beta_dist (double x, double a, double b, double tol, double *p, double *q, double *pdf, NagError *fail)

## 3  Description

The probability density function of the beta distribution with parameters $a$ and $b$ is:
 $fB:a,b=Γa+b ΓaΓb Ba-11-Bb-1, 0≤B≤1;a,b>0.$
The lower tail probability, $P\left(B\le \beta :a,b\right)$ is defined by
 $PB≤β:a,b=Γa+b ΓaΓb ∫0βBa-11-Bb-1dB=Iβa,b, 0≤β≤1;a,b>0.$
The function ${I}_{x}\left(a,b\right)$, also known as the incomplete beta function is calculated using nag_incomplete_beta (s14ccc).

## 4  References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5  Arguments

1:     xdoubleInput
On entry: $\beta$, the value of the beta variate.
Constraint: $0.0\le {\mathbf{x}}\le 1.0$.
On entry: $a$, the first parameter of the required beta distribution.
Constraint: $0.0<{\mathbf{a}}\le {10}^{6}$.
3:     bdoubleInput
On entry: $b$, the second parameter of the required beta distribution.
Constraint: $0.0<{\mathbf{b}}\le {10}^{6}$.
4:     toldoubleInput
On entry: this argument is no longer referenced, but is included for backwards compatability.
5:     pdouble *Output
On exit: the lower tail probability, $P\left(B\le \beta :a,b\right)$.
6:     qdouble *Output
On exit: the upper tail probability, $P\left(B\ge \beta :a,b\right)$.
7:     pdfdouble *Output
On exit: the probability density function, $f\left(B:a,b\right)$.
8:     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.
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.
NE_REAL_ARG_GT
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\le 1.0$.
NE_REAL_ARG_LE
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{b}}>0.0$.
NE_REAL_ARG_LT
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\ge 0.0$.

## 7  Accuracy

The accuracy is limited by the error in the incomplete beta function. See Section 7 in nag_incomplete_beta (s14ccc) for further details.

Not applicable.

None.

## 10  Example

This example reads values from a number of beta distributions and computes the associated upper and lower tail probabilities and the corresponding value of the probability density function.

### 10.1  Program Text

Program Text (g01eece.c)

### 10.2  Program Data

Program Data (g01eece.d)

### 10.3  Program Results

Program Results (g01eece.r)