naginterfaces.library.rand.dist_​cauchy

naginterfaces.library.rand.dist_cauchy(n, xmed, semiqr, statecomm)

dist_cauchy generates a vector of pseudorandom numbers from a Cauchy distribution with median $a$ and semi-interquartile range $b$.

For full information please refer to the NAG Library document for g05sc

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g05/g05scf.html

Parameters

nint

$n$, the number of pseudorandom numbers to be generated.

xmedfloat

$a$, the median of the distribution.

semiqrfloat

$b$, the semi-interquartile range of the distribution.

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

Returns

xfloat, ndarray, shape $\left(n\right)$

The $n$ pseudorandom numbers from the specified Cauchy distribution.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $3$)

On entry, $semiqr=⟨value⟩$.

Constraint: $semiqr\ge 0.0$.

(errno $4$)

On entry, $statecomm$[‘state’] vector has been corrupted or not initialized.

Notes

The distribution has PDF (probability density function)

$$f\left(x\right)=\frac{1}{\pi b(1+{\left(\frac{x-a}{b}\right)}^2)}\text{.}$$

dist_cauchy returns the value

$$a+b\frac{2y_1-1}{y_2}\text{,}$$

where $y_1$ and $y_2$ are a pair of consecutive pseudorandom numbers from a uniform distribution over $(0,1)$, such that

$${(2y_1-1)}^2+y_2^2\le 1\text{.}$$

One of the initialization functions init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to dist_cauchy.

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