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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_dist_f (g05sh)

Purpose

nag_rand_dist_f (g05sh) generates a vector of pseudorandom numbers taken from an F$F$ (or Fisher's variance ratio) distribution with μ$\mu$ and ν$\nu$ degrees of freedom.

Syntax

[state, x, ifail] = g05sh(n, df1, df2, state)
[state, x, ifail] = nag_rand_dist_f(n, df1, df2, state)

Description

The distribution has PDF (probability density function)
 f (x) = ( (( μ + ν − 2 )/2) ! x (1/2) μ − 1 )/( ((1/2)μ − 1) ! ((1/2)ν − 1) ! (1 + μ/νx) (1/2) (μ + ν) ) × (μ/ν)(1/2)μ if ​ x > 0 , f(x) = 0 otherwise.
$f (x) = ( μ+ν-2 2 ) ! x 12 μ-1 ( 12 μ-1)! (12 ν-1 ) ! ( 1+ μν x ) 12 (μ+ν) × (μν) 12μ if ​ x>0 , f(x)=0 otherwise.$
nag_rand_dist_f (g05sh) calculates the values
 (ν yi)/(μ zi) ,   i = 1,2, … ,n , $ν yi μ zi , i=1,2,…,n ,$
where yi${y}_{i}$ and zi${z}_{i}$ are generated by nag_rand_dist_gamma (g05sj) from gamma distributions with parameters ((1/2)μ,2)$\left(\frac{1}{2}\mu ,2\right)$ and ((1/2)ν,2)$\left(\frac{1}{2}\nu ,2\right)$ respectively (i.e., from χ2${\chi }^{2}$-distributions with μ$\mu$ and ν$\nu$ degrees of freedom).
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_dist_f (g05sh).

References

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

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
n$n$, the number of pseudorandom numbers to be generated.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     df1 – int64int32nag_int scalar
μ$\mu$, the number of degrees of freedom of the distribution.
Constraint: df11${\mathbf{df1}}\ge 1$.
3:     df2 – int64int32nag_int scalar
ν$\nu$, the number of degrees of freedom of the distribution.
Constraint: df21${\mathbf{df2}}\ge 1$.
4:     state( : $:$) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

None.

None.

Output Parameters

1:     state( : $:$) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains updated information on the state of the generator.
2:     x(n) – double array
The n$n$ pseudorandom numbers from the specified F$F$-distribution.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, n < 0${\mathbf{n}}<0$.
ifail = 2${\mathbf{ifail}}=2$
On entry, df1 < 1${\mathbf{df1}}<1$.
ifail = 3${\mathbf{ifail}}=3$
On entry, df2 < 1${\mathbf{df2}}<1$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, state vector was not initialized or has been corrupted.

Accuracy

Not applicable.

The time taken by nag_rand_dist_f (g05sh) increases with μ$\mu$ and ν$\nu$.

Example

```function nag_rand_dist_f_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
df1 = int64(2);
df2 = int64(3);
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_f(n, df1, df2, state)
```
```

state =

17
1234
1
0
18942
20099
15825
20302
17917
13895
19930
8
0
1234
1
1
1234

x =

1.4401
1.8083
0.3638
0.5464
4.0895

ifail =

0

```
```function g05sh_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
df1 = int64(2);
df2 = int64(3);
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sh(n, df1, df2, state)
```
```

state =

17
1234
1
0
18942
20099
15825
20302
17917
13895
19930
8
0
1234
1
1
1234

x =

1.4401
1.8083
0.3638
0.5464
4.0895

ifail =

0

```