Voigt Profile Distribution

Voigt Profile Distribution#

class ctx.dist_voigt_profile(n1, n2, lambda, **kwargs)#

The Voigt profile is a probability distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data from spectroscopy or diffraction.

These functions return PDF, CDF, and ICDF of the Voigt profile distribution with location \(a\), scale \(b > 0\), and the support interval \((-\infty,+\infty)\) :

See also: Wikipedia [1312].

dist_voigt_profile.pdf(x)#

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following an Voigt profile distribution:

\[\text{pdf}_X(x) = V(x; \sigma, \gamma) = \frac{\Re[w(z)]}{\sigma \sqrt{2 \pi}}, \quad \text{where } z = \frac{x + i\gamma}{\sigma \sqrt{\pi}}.\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", dist_voigt_profile(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_voigt_profile.cdf(x)#

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an Voigt profile distribution:

\[\text{cdf}_X(x) = \Re \left[ \frac{1}{2} + \frac{\text{erf}(z)}{2} + \frac{i z^2}{\pi} {}_2F_2 \left( 1,1; \tfrac{3}{2}, 2; -z^2 \right) \right].\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", dist_voigt_profile(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_voigt_profile.sf(x)#

Returns \(\text{sf}_X(x)\), the survival function function (sf) of a random variable \(X\), following an Voigt profile distribution:

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t.\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", dist_voigt_profile(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_voigt_profile.qtf(q)#

Returns \(\text{qtf}_X(x)\), the quantile function function (qtf) of a random variable \(X\), following an Voigt profile distribution:

\[\text{qtf}_X(q) = \text{no closed form}.\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", dist_voigt_profile(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_voigt_profile.isf(q)#

Returns \(\text{isf}_X(q)\), the inverse survival function function (isf) of a random variable \(X\), following an Voigt profile distribution:

\[\text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q).\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", dist_voigt_profile(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_voigt_profile.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following an Voigt profile distribution:

\[C_X(t) = e^{-\gamma |t| -\sigma^2 t^2 /2}.\]
>>> from mpdistrib import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", dist_voigt_profile(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_voigt_profile.m_x(t)#

Returns NaN, since the moment generating function does not exist.

dist_voigt_profile.k_x(t, k=0)#

Returns NaN, since the cumulant generating function does not exist.

dist_voigt_profile.moments(k)#

Returns NaN, since moments do not exist.

dist_voigt_profile.cumulants(k)#

Returns NaN, since cumulants do not exist.

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