naginterfaces.library.rand.dist_​students_​t

naginterfaces.library.rand.dist_students_t(n, df, statecomm)

dist_students_t generates a vector of pseudorandom numbers taken from a Student’s $t$-distribution with $\nu$ degrees of freedom.

For full information please refer to the NAG Library document for g05sn

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g05/g05snf.html

Parameters

nint

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

dfint

$\nu$, the number of degrees of freedom 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 Student’s $t$-distribution.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $2$)

On entry, $df=⟨value⟩$.

Constraint: $df\ge 1$.

(errno $3$)

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

Notes

The distribution has PDF (probability density function)

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

dist_students_t calculates the values

$$y_i\sqrt{\frac{\nu }{z_i}}\text{,}i=1,\dots ,n$$

where the $y_i$ are generated by dist_normal() from a Normal distribution with mean $0$ and variance $1.0$, and the $z_i$ are generated by dist_gamma() from a gamma distribution with parameters $\frac{1}{2}\nu$ and $2$ (i.e., from a ${\chi }^2$-distribution with $\nu$ degrees of freedom).

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_students_t.

References

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