Boost: Cauchy distribution

Boost: Cauchy distribution#

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

See also Wikipedia [1239], MathWorld [890], BoostMath [58], Ehrhardt [309] (3.9.4).

Ctx.cauchy_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 Cauchy distribution:

\[\text{pdf}(x) = \frac{1}{\pi b (1+((x-a)/b)^2)}.\]

The following example shows both forms of the syntax:

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

Ctx.cauchy_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 Cauchy distribution:

\[\text{cdf}(x) = \frac{1}{2} + \frac{1}{\pi} \arctan \left(\frac{x-a}{b} \right).\]

The following example shows both forms of the syntax:

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

Ctx.cauchy_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 Cauchy distribution:

\[\begin{split}\text{qtf}(q) =\begin{cases} a-b/\tan(\pi q), & q<0.5,\\ a, & q=0.5,\\ a-b/\tan(\pi (1-q)) & q>0.5. \end{cases}\end{split}\]

The following example shows both forms of the syntax:

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

class ctx.dist_cauchy(a, b)#

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

The Cauchy distribution is a continuous probability distribution with parameters \(a \in \mathbb{R}\) (location), \(b > 0\) (scale), and the support interval \((-\infty, +\infty)\). See also Wikipedia [1239], MathWorld [890], BoostMath [58], R (Statistical System) [546].

dist_cauchy.pdf(x)#

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

\[\text{pdf}_X(x) = \frac{1}{\pi(1+((x-a)/b)^2)}.\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", cauchy(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_cauchy.cdf(x)#

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

\[\text{cdf}_X(x) = \frac{1}{2} + \frac{1}{\pi} \arctan \left(\frac{x-a}{b} \right).\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", cauchy(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_cauchy.sf(x)#

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

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \frac{1}{2} - \frac{1}{\pi} \arctan \left(\frac{x-a}{b} \right).\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", cauchy(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_cauchy.qtf(q)#

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

\[\begin{split}\text{qtf}_X(q) =\begin{cases} a-b/\tan(\pi q), & q<0.5,\\ a, & q=0.5,\\ a-b/\tan(\pi (1-q)) & q>0.5. \end{cases}\end{split}\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", cauchy(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_cauchy.isf(q)#

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

\[\begin{split}\text{isf}_X(q) =\begin{cases} a+b/\tan(\pi q), & q<0.5,\\ a, & q=0.5,\\ a+b/\tan(\pi (1-q)) & q>0.5. \end{cases}\end{split}\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", cauchy(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_cauchy.c_x(t)#

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

\[C_X(t) = \exp(a \cdot i t - b |t|).\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", cauchy(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00
dist_cauchy.m_x(t)#

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

dist_cauchy.k_x(t, k=0)#

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

dist_cauchy.moments(k)#

Returns NaN, since moments do not exist.

dist_cauchy.cumulants(k)#

Returns NaN, since cumulants do not exist.

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