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# NAG Toolbox: nag_rand_dist_chisq (g05sd)

## Purpose

nag_rand_dist_chisq (g05sd) generates a vector of pseudorandom numbers taken from a ${\chi }^{2}$-distribution with $\nu$ degrees of freedom.

## Syntax

[state, x, ifail] = g05sd(n, df, state)
[state, x, ifail] = nag_rand_dist_chisq(n, df, state)

## Description

The distribution has PDF (probability density function)
 $fx = x ν/2-1 × e -x/2 2 ν/2 × ν/2-1 ! if ​ x>0 ; fx=0 otherwise.$
This is the same as a gamma distribution with parameters $\nu /2$ and $2$.
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_chisq (g05sd).

## References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{df}$int64int32nag_int scalar
$\nu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df}}\ge 1$.
3:     $\mathrm{state}\left(:\right)$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.

### Output Parameters

1:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Contains updated information on the state of the generator.
2:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The $n$ pseudorandom numbers from the specified ${\chi }^{2}$-distribution.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{df}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

The time taken by nag_rand_dist_chisq (g05sd) increases with $\nu$.

## Example

This example prints five pseudorandom numbers from a ${\chi }^{2}$-distribution with five degrees of freedom, generated by a single call to nag_rand_dist_chisq (g05sd), after initialization by nag_rand_init_repeat (g05kf).
```function g05sd_example

fprintf('g05sd example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
genid, subid, seed);

% Number of variates
n = int64(5);

% Parameters
df = int64(5);

% Generate variates from Chi^2 distribution
[state, x, ifail] = g05sd( ...
n, df, state);

disp('Variates');
disp(x);

```
```g05sd example results

Variates
4.4731
5.9371
1.7636
2.9812
4.3280

```

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Chapter Contents
Chapter Introduction
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