!!!Boost: Moyal Distribution

The following functions return the pdf, cdf, qtf or boost class of the Moyal distribution with location \(a \in \mathbb{R}\), scale \(b > 0\), and the support interval \((-\infty, +\infty)\).

See also MathWorld [261], G. [356], Walck [863], Ehrhardt [309] (3.9.21).

Ctx.moyal_pdf(x, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the Moyal distribution:

\[\text{pdf}(x) = \frac{1}{\sqrt{2\pi}b} \exp\left( -\frac{x-a}{2b} -\frac{1}{2} e^{-\frac{x-a}{b}} \right).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("MoyalPdf(x, a, b): ", MoyalPdf(x, a, b))
>>> print ("dist_moyal(a, b).pdf(x): ", dist_moyal(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.moyal_cdf(x, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the Moyal distribution:

\[\text{cdf}(x) = 1 - P \left( \frac{1}{2}, \frac{e^{-x}}{2} \right) = \text{erfc}\left( \frac{e^{-\frac{x-a}{2b}}}{\sqrt{2}} \right).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("MoyalCdf(x, a, b): ", MoyalCdf(x, a, b))
>>> print ("dist_moyal(a, b).cdf(x): ", dist_moyal(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.moyal_qtf(q, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the Moyal distribution:

\[\text{qtf}(q) = a - b \log\left( 2 (\text{erfc}^{-1}(q))^2 \right)\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("MoyalQtf(q, a, b): ", MoyalQtf(q, a, b))
>>> print ("dist_moyal(a, b).qtf(q): ", dist_moyal(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_moyal(a, b)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

The Moyal distribution is a continuous probability distribution with location \(a \in \mathbb{R}\), scale \(b > 0\), and the support interval \((-\infty, +\infty)\). See also MathWorld [261], G. [356], Walck [863].

Other References: Amath: inversion, Walck: moments and char func.

dist_moyal.pdf(x)

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

\[\text{pdf}_X(x) = \frac{1}{\sqrt{2\pi}b} \exp\left( -\frac{x-a}{2b} -\frac{1}{2} e^{-\frac{x-a}{b}} \right).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", moyal(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_moyal.cdf(x)

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

\[\text{cdf}_X(x) = 1 - P \left( \frac{1}{2}, \frac{e^{-x}}{2} \right) = \text{erfc}\left( \frac{e^{-\frac{x-a}{2b}}}{\sqrt{2}} \right).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", moyal(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_moyal.sf(x)

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

\[\text{sf}_X(x) = P \left( \frac{1}{2}, \frac{e^{-x}}{2} \right) = \text{erf}\left( \frac{e^{-\frac{x-a}{2b}}}{\sqrt{2}} \right).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", moyal(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_moyal.qtf(q)

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

\[\text{qtf}_X(q) = a - b \log\left( 2 (\text{erfc}^{-1}(q))^2 \right)\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", moyal(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_moyal.isf(q)

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

\[\text{isf}_X(q) = a - b \log\left( 2 (\text{erf}^{-1}(q))^2 \right)\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", moyal(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_moyal.c_x(t)

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

\[C_X(t) = \frac{2^{it}}{\sqrt{\pi}} \Gamma\left( \tfrac{1}{2} - it \right).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", moyal(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_moyal.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following an Moyal distribution:

\[M_X(t) = \frac{2^{t}}{\sqrt{\pi}} \Gamma\left( \tfrac{1}{2} - t \right).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("m_x: ", moyal(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_moyal.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following an Moyal distribution:

\[K_X(t) = \log\left[ \frac{2^{t}}{\sqrt{\pi}} \Gamma\left( \tfrac{1}{2} - t \right) \right].\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", moyal(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_moyal.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Moyal distribution. The moments are calculated from the cumulants.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", moyal(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_moyal.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following an Moyal distribution. The cumulants are calculated from the moments.

\[\kappa_{X}(1) = \log(2) + \gamma, \qquad \kappa_{X}(n) = (n-1)! (2^n -1) \zeta(n), \quad n \ge 2.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", moyal(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00