freealg.distributions.Meixner

class freealg.distributions.Meixner(a, b, c)

Meixner distribution.

Parameters:

afloat

Parameter \(a\) of the distribution. See Notes.

bfloat

Parameter \(b\) of the distribution. See Notes.

Notes

The Meixner distribution has the absolutely-continuous density

\[\mathrm{d} \rho(x) = \frac{4(1+b) - (x-a)^2}{2 \pi (b x^2 + a x + 1)} \mathbf{1}_{x \in [\lambda_{-}, \lambda_{+}]} \mathrm{d}{x}\]

where \(a, b\) are the shape parameters of the distribution. The edges of the support are

\[\lambda_{\pm} = a \pm 2 \sqrt{1 + b}.\]

References

[1]

Saitoh, N. & Yosnida, M. (2001). The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory. Probab. Math. Statist., 21, 159–170.

Examples

>>> from freealg.distributions import Meixner
>>> mx = Meixner(2, 3)

Methods

density([x, plot, latex, save, eig])	Density of distribution.
hilbert([x, plot, latex, save])	Hilbert transform of the distribution.
stieltjes([x, y, plot, on_disk, latex, save])	Stieltjes transform of distribution.
sample(size[, x_min, x_max, method, seed, ...])	Sample from distribution.
matrix(size[, seed])	Generate matrix with the spectral density of the distribution.