g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rand_f (g05shc)

## 1  Purpose

nag_rand_f (g05shc) generates a vector of pseudorandom numbers taken from an $F$ (or Fisher's variance ratio) distribution with $\mu$ and $\nu$ degrees of freedom.

## 2  Specification

 #include #include
 void nag_rand_f (Integer n, Integer df1, Integer df2, Integer state[], double x[], NagError *fail)

## 3  Description

The distribution has PDF (probability density function)
 $f x = μ+ν-2 2 ! x 12 μ-1 12 μ-1! 12 ν-1 ! 1+ μν x 12 μ+ν × μν 12μ if ​ x>0 , fx=0 otherwise.$
nag_rand_f (g05shc) calculates the values
 $ν yi μ zi , i=1,2,…,n ,$
where ${y}_{i}$ and ${z}_{i}$ are generated by nag_rand_gamma (g05sjc) from gamma distributions with parameters $\left(\frac{1}{2}\mu ,2\right)$ and $\left(\frac{1}{2}\nu ,2\right)$ respectively (i.e., from ${\chi }^{2}$-distributions with $\mu$ and $\nu$ degrees of freedom).
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_f (g05shc).

## 4  References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     df1IntegerInput
On entry: $\mu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df1}}\ge 1$.
3:     df2IntegerInput
On entry: $\nu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df2}}\ge 1$.
4:     state[$\mathit{dim}$]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions 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:     x[n]doubleOutput
On exit: the $n$ pseudorandom numbers from the specified $F$-distribution.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{df1}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df1}}\ge 1$.
On entry, ${\mathbf{df2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df2}}\ge 1$.
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_INVALID_STATE
On entry, state vector has been corrupted or not initialized.

## 7  Accuracy

Not applicable.

The time taken by nag_rand_f (g05shc) increases with $\mu$ and $\nu$.

## 9  Example

This example prints five pseudorandom numbers from an $F$-distribution with two and three degrees of freedom, generated by a single call to nag_rand_f (g05shc), after initialization by nag_rand_init_repeatable (g05kfc).

### 9.1  Program Text

Program Text (g05shce.c)

None.

### 9.3  Program Results

Program Results (g05shce.r)