!!!Boost: Maxwell Distribution

Returns the pdf, cdf, qtf or boost class of a random variable \(X\), following a Maxwell distribution with scale \(b > 0\), and the support interval \((0,+\infty)\).

See also Wikipedia [1278], MathWorld [904], Ehrhardt [309] (3.9.20).

Ctx.maxwell_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 Maxwell distribution:

\[\text{pdf}(x) = \sqrt{\frac{2}{\pi}} \frac{x^2}{b^3} \exp\left( -\frac{x^2}{2b^2} \right).\]

The following example shows both forms of the syntax:

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

Ctx.maxwell_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 Maxwell distribution:

\[\text{cdf}(x) = P\left( \frac{3}{2}, \frac{x^2}{2b^2} \right).\]

Here \(P(\cdot)\) denotes the lower regularized incomplete gamma function (RealGammaP).

The following example shows both forms of the syntax:

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

Ctx.maxwell_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 Maxwell distribution:

\[\text{qtf}(q) = b \sqrt{2 P^{-1}\left(\frac{3}{2}, q\right)}.\]

Here \(P^{-1}(\cdot)\) denotes the inverse of the lower regularized incomplete gamma function (RealGammaPInv).

The following example shows both forms of the syntax:

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

class ctx.dist_maxwell(b)

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

The Maxwell distribution is a continuous probability distribution with scale \(b > 0\), and the support interval \((0,+\infty)\). See also Wikipedia [1278], MathWorld [904], Witkovský [1639].

dist_maxwell.pdf(x)

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

\[\text{pdf}_X(x) = \sqrt{\frac{2}{\pi}} \frac{x^2}{b^3} \exp\left( -\frac{x^2}{2b^2} \right).\]

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

dist_maxwell.cdf(x)

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

\[\text{cdf}_X(x) = P\left( \frac{3}{2}, \frac{x^2}{2b^2} \right).\]

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

dist_maxwell.sf(x)

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

\[\text{sf}_X(x) = Q\left( \frac{3}{2}, \frac{x^2}{2b^2} \right).\]

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

dist_maxwell.qtf(q)

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

\[\text{qtf}_X(q) = b \sqrt{2 P^{-1}\left(\frac{3}{2}, q\right)}.\]

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

dist_maxwell.isf(q)

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

\[\text{isf}_X(q) = b \sqrt{2 Q^{-1}\left(\frac{3}{2}, q\right)}.\]

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

dist_maxwell.c_x(t)

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

\[C_X(t) = M\left( \frac{3}{2}, \frac{1}{2} \frac{-t^2}{2} \right) + \frac{2 \sqrt{2} \, i t }{\sqrt{\pi}} M\left(2, \frac{3}{2}, \frac{-t^2}{2} \right).\]

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

dist_maxwell.m_x(t)

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

\[M_X(t) = M\left( \frac{3}{2}, \frac{1}{2} \frac{t^2}{2} \right) + \frac{2 \sqrt{2} \, t }{\sqrt{\pi}} M\left( 2, \frac{3}{2}, \frac{t^2}{2} \right).\]

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

dist_maxwell.k_x(t, k=0)

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

\[K_X(t) = \log \left[ M\left( \frac{3}{2}, \frac{1}{2} \frac{t^2}{2} \right) + \frac{2 \sqrt{2} \, t }{\sqrt{\pi}} M\left(2, \frac{3}{2}, \frac{t^2}{2} \right) \right].\]

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

dist_maxwell.moments(k)

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

\[\begin{split}\mu_{X}(r) = \begin{cases} \sqrt{\frac{2}{\pi}} k! \alpha^{2k-1} & \text{for } n=2k-1,\\ (n+1)!! \alpha^n & \text{for } n \text{ even}, \end{cases}\end{split}\]

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

dist_maxwell.cumulants(k)

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

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