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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_dist_chisq (g05sd)

## Purpose

nag_rand_dist_chisq (g05sd) generates a vector of pseudorandom numbers taken from a χ2${\chi }^{2}$-distribution with ν$\nu$ 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)
 f(x) = ( xν / 2 − 1 × e − x / 2 )/( 2ν / 2 × (ν / 2 − 1) ! ) if ​ x > 0 ; f(x) = 0 otherwise.
$f(x) = x ν/2-1 × e -x/2 2 ν/2 × ( ν/2-1 ) ! if ​ x>0 ; f(x)=0 otherwise.$
This is the same as a gamma distribution with parameters ν / 2$\nu /2$ and 2$2$.
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:     n – int64int32nag_int scalar
n$n$, the number of pseudorandom numbers to be generated.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     df – int64int32nag_int scalar
ν$\nu$, the number of degrees of freedom of the distribution.
Constraint: df1 ${\mathbf{df}}\ge 1$.
3:     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 χ2${\chi }^{2}$-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, df < 1${\mathbf{df}}<1$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, state vector was not initialized or has been corrupted.

## Accuracy

Not applicable.

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

## Example

function nag_rand_dist_chisq_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
df = int64(5);
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_chisq(n, df, state)

state =

17
1234
1
0
3990
775
3088
31015
17917
13895
19930
8
0
1234
1
1
1234

x =

4.4731
5.9371
1.7636
2.9812
4.3280

ifail =

0

function g05sd_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
df = int64(5);
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sd(n, df, state)

state =

17
1234
1
0
3990
775
3088
31015
17917
13895
19930
8
0
1234
1
1
1234

x =

4.4731
5.9371
1.7636
2.9812
4.3280

ifail =

0