Boost: Rayleigh distribution

The following functions return the pdf, cdf, qtf or boost class of the Rayleigh distribution with scale \(b > 0\) and the support interval \((0, +\infty)\).

See also Wikipedia [1282], MathWorld [909], BoostMath [91], Ehrhardt [309] (3.9.27).

Ctx.rayleigh_pdf(x, b)

where Ctx is Math53 or CtxBoost.

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

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

The following example shows both forms of the syntax:

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

Ctx.rayleigh_cdf(x, b)

where Ctx is Math53 or CtxBoost.

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

\[\text{cdf}(x) = 1 - \exp \left(- \frac{x^2}{2b^2}\right) = -\text{expm1} \left(- \frac{x^2}{2b^2}\right).\]

The following example shows both forms of the syntax:

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

Ctx.rayleigh_qtf(q, b)

where Ctx is Math53 or CtxBoost.

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

\[\text{qtf}(q) = b \sqrt{-2 \cdot \text{log1p}(-q)}.\]

The following example shows both forms of the syntax:

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

class ctx.dist_rayleigh(b)

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

The Rayleigh distribution is a continuous probability distribution with scale \(b > 0\) and the support interval \((0, +\infty)\). See also Wikipedia [1282], MathWorld [909], BoostMath [91], Witkovský [1628].

dist_rayleigh.pdf(x)

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

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

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

dist_rayleigh.cdf(x)

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

\[\text{cdf}_X(x) = 1 - \exp \left(- \frac{x^2}{2b^2}\right) = -\text{expm1} \left(- \frac{x^2}{2b^2}\right).\]

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

dist_rayleigh.sf(x)

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

\[\text{sf}_X(x) = \exp \left(- \frac{x^2}{2b^2}\right).\]

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

dist_rayleigh.qtf(q)

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

\[\text{qtf}_X(q) = b \sqrt{-2 \cdot \text{log1p}(-q)}.\]

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

dist_rayleigh.isf(q)

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

\[\text{isf}_X(q) = b \sqrt{-2 \cdot \log(q)}.\]

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

dist_rayleigh.c_x(t)

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

\[C_X(t) = 1 + b t e^{b^2 t^2 /2} \sqrt{\frac{\pi}{2}} \left(\text{erfi} \left( \frac{bt}{\sqrt{2}} \right) -i \right).\]

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

dist_rayleigh.m_x(t)

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

\[M_X(t) = 1 + b t e^{b^2 t^2 /2} \sqrt{\frac{\pi}{2}} \left(\text{erf} \left( \frac{bt}{\sqrt{2}} \right) +1 \right).\]

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

dist_rayleigh.k_x(t, k=0)

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

\[K_X(t) = K_X(t) = \log(M_X(t)).\]

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

dist_rayleigh.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Rayleigh distribution.

\[\mu'_{X}(j) = b^j 2^{j/2} \Gamma(1+j/2).\]

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

dist_rayleigh.cumulants(k)

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

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