# NAG C Library Function Document

## 1Purpose

nag_rand_beta (g05sbc) generates a vector of pseudorandom numbers taken from a beta distribution with parameters $a$ and $b$.

## 2Specification

 #include #include
 void nag_rand_beta (Integer n, double a, double b, Integer state[], double x[], NagError *fail)

## 3Description

The beta distribution has PDF (probability density function)
One of four algorithms is used to generate the variates depending on the values of $a$ and $b$. Let $\alpha$ be the maximum and $\beta$ be the minimum of $a$ and $b$. Then the algorithms are as follows:
 (i) if $\alpha <0.5$, Johnk's algorithm is used, see for example Dagpunar (1988). This generates the beta variate as ${u}_{1}^{1/a}/\left(\begin{array}{c}{u}_{1}^{1/a}+{u}_{2}^{1/b}\end{array}\right)$, where ${u}_{1}$ and ${u}_{2}$ are uniformly distributed random variates; (ii) if $\beta >1$, the algorithm BB given by Cheng (1978) is used. This involves the generation of an observation from a beta distribution of the second kind by the envelope rejection method using a log-logistic target distribution and then transforming it to a beta variate; (iii) if $\alpha >1$ and $\beta <1$, the switching algorithm given by Atkinson (1979) is used. The two target distributions used are ${f}_{1}\left(x\right)=\beta {x}^{\beta }$ and ${f}_{2}\left(x\right)=\alpha {\left(1-x\right)}^{\beta -1}$, along with the approximation to the switching parameter of $t=\left(1-\beta \right)/\left(\alpha +1-\beta \right)$; (iv) in all other cases, Cheng's BC algorithm (see Cheng (1978)) is used with modifications suggested by Dagpunar (1988). This algorithm is similar to BB, used when $\beta >1$, but is tuned for small values of $a$ and $b$.
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_beta (g05sbc).

## 4References

Atkinson A C (1979) A family of switching algorithms for the computer generation of beta random variates Biometrika 66 141–5
Cheng R C H (1978) Generating beta variates with nonintegral shape parameters Comm. ACM 21 317–322
Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{a}$doubleInput
On entry: $a$, the parameter of the beta distribution.
Constraint: ${\mathbf{a}}>0.0$.
3:    $\mathbf{b}$doubleInput
On entry: $b$, the parameter of the beta distribution.
Constraint: ${\mathbf{b}}>0.0$.
4:    $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
5:    $\mathbf{x}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the $n$ pseudorandom numbers from the specified beta distribution.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>0.0$.

Not applicable.

## 8Parallelism and Performance

nag_rand_beta (g05sbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

To generate an observation, $y$, from the beta distribution of the second kind from an observation, $x$, generated by nag_rand_beta (g05sbc) the transformation, $y=x/\left(1-x\right)$, may be used.

## 10Example

This example prints a set of five pseudorandom numbers from a beta distribution with parameters $a=2.0$ and $b=2.0$, generated by a single call to nag_rand_beta (g05sbc), after initialization by nag_rand_init_repeatable (g05kfc).

### 10.1Program Text

Program Text (g05sbce.c)

None.

### 10.3Program Results

Program Results (g05sbce.r)