# NAG CL Interfaceg01sec (prob_​beta_​vector)

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## 1Purpose

g01sec computes a number of lower or upper tail probabilities for the beta distribution.

## 2Specification

 #include
 void g01sec (Integer ltail, const Nag_TailProbability tail[], Integer lbeta, const double beta[], Integer la, const double a[], Integer lb, const double b[], double p[], Integer ivalid[], NagError *fail)
The function may be called by the names: g01sec, nag_stat_prob_beta_vector or nag_prob_beta_vector.

## 3Description

The lower tail probability, $P\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$ is defined by
 $P( Bi ≤ βi :ai,bi) = Γ (ai+bi) Γ (ai) Γ (bi) ∫ 0 βi Bi ai-1 (1-Bi) bi-1 dBi = Iβi (ai,bi) , 0 ≤ βi ≤ 1 ; ai , bi > 0 .$
The function ${I}_{{\beta }_{i}}\left({a}_{i},{b}_{i}\right)$, also known as the incomplete beta function is calculated using s14ccc.
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.
NIST Digital Library of Mathematical Functions
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Majumder K L and Bhattacharjee G P (1973) Algorithm AS 63. The incomplete beta integral Appl. Statist. 22 409–411

## 5Arguments

1: $\mathbf{ltail}$Integer Input
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2: $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbability Input
On entry: indicates whether a lower or upper tail probabilities are required. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lbeta}},{\mathbf{la}},{\mathbf{lb}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({B}_{i}\ge {\beta }_{i}:{a}_{i},{b}_{i}\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3: $\mathbf{lbeta}$Integer Input
On entry: the length of the array beta.
Constraint: ${\mathbf{lbeta}}>0$.
4: $\mathbf{beta}\left[{\mathbf{lbeta}}\right]$const double Input
On entry: ${\beta }_{i}$, the value of the beta variate with ${\beta }_{i}={\mathbf{beta}}\left[j\right]$, .
Constraint: $0.0\le {\mathbf{beta}}\left[\mathit{j}-1\right]\le 1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lbeta}}$.
5: $\mathbf{la}$Integer Input
On entry: the length of the array a.
Constraint: ${\mathbf{la}}>0$.
6: $\mathbf{a}\left[{\mathbf{la}}\right]$const double Input
On entry: ${a}_{i}$, the first parameter of the required beta distribution with ${a}_{i}={\mathbf{a}}\left[j\right]$, .
Constraint: ${\mathbf{a}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
7: $\mathbf{lb}$Integer Input
On entry: the length of the array b.
Constraint: ${\mathbf{lb}}>0$.
8: $\mathbf{b}\left[{\mathbf{lb}}\right]$const double Input
On entry: ${b}_{i}$, the second parameter of the required beta distribution with ${b}_{i}={\mathbf{b}}\left[j\right]$, .
Constraint: ${\mathbf{b}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
9: $\mathbf{p}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lbeta}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: ${p}_{i}$, the probabilities for the beta distribution.
10: $\mathbf{ivalid}\left[\mathit{dim}\right]$Integer Output
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lbeta}},{\mathbf{la}},{\mathbf{lb}}\right)$.
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, invalid value supplied in tail when calculating ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
On entry, ${\beta }_{i}<0.0$, or, ${\beta }_{i}>1.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
On entry, ${a}_{i}\le 0.0$, or, ${b}_{i}\le 0.0$.
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{la}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lb}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lbeta}}>0$.
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ltail}}>0$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of beta, a, b or tail was invalid.

## 7Accuracy

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

## 8Parallelism and Performance

g01sec is not threaded in any implementation.

None.

## 10Example

This example reads values from a number of beta distributions and computes the associated lower tail probabilities.

### 10.1Program Text

Program Text (g01sece.c)

### 10.2Program Data

Program Data (g01sece.d)

### 10.3Program Results

Program Results (g01sece.r)