.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| .. _rst_dist_cauchy: 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 :cite:p:`WikipediaDis11`, MathWorld :cite:p:`WolframDis11`, BoostMath :cite:p:`BoostDis11`, :cite:t:`Ehrhardt2018` (3.9.4). |cr| .. _Ctx_CauchyPdf: .. method:: 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 (:ref:`Pdf `) of the Cauchy distribution: .. math:: \text{pdf}(x) = \frac{1}{\pi b (1+((x-a)/b)^2)}. The following example shows both forms of the syntax: .. code-block:: pycon >>> 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 |cr| .. _Ctx_CauchyCdf: .. method:: 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 (:ref:`Cdf `) of the Cauchy distribution: .. math:: \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: .. code-block:: pycon >>> 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 |cr| .. _Ctx_CauchyQtf: .. method:: Ctx.cauchy_qtf(q, a=0, b=1) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{qtf}(q)`, the value of the quantile function (:ref:`Qtf `) of the Cauchy distribution: .. math:: \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} The following example shows both forms of the syntax: .. code-block:: pycon >>> 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 |cr| .. py: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 :cite:p:`WikipediaDis11`, MathWorld :cite:p:`WolframDis11`, BoostMath :cite:p:`BoostDis11`, R (Statistical System) :cite:p:`RDis11`. |cr| .. method:: dist_cauchy.pdf(x) Returns `\text{pdf}_X(x)`, the probability density function (pdf) of a random variable `X`, following a Cauchy distribution: .. math:: \text{pdf}_X(x) = \frac{1}{\pi(1+((x-a)/b)^2)}. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("pdf: ", cauchy(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_cauchy.cdf(x) Returns `\text{cdf}_X(x)`, the cumulative distribution function (cdf) of a random variable `X`, following a Cauchy distribution: .. math:: \text{cdf}_X(x) = \frac{1}{2} + \frac{1}{\pi} \arctan \left(\frac{x-a}{b} \right). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("cdf: ", cauchy(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_cauchy.sf(x) Returns `\text{sf}_X(x)`, the survival function (sf) of a random variable `X`, following a Cauchy distribution: .. math:: \text{sf}_X(x) = 1 - \text{cdf}_X(x) = \frac{1}{2} - \frac{1}{\pi} \arctan \left(\frac{x-a}{b} \right). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print (" sf: ", cauchy(mu, sigma).pdf(x)) sf: 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_cauchy.qtf(q) Returns `\text{qtf}_X(x)`, the quantile function (qtf) of a random variable `X`, following a Cauchy distribution: .. math:: \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} .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("qtf: ", cauchy(mu, sigma).qtf(q)) qtf: 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_cauchy.isf(q) Returns `\text{isf}_X(q)`, the inverse survival function (isf) of a random variable `X`, following a Cauchy distribution: .. math:: \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} .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("isf: ", cauchy(mu, sigma).isf(q)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_cauchy.c_x(t) Returns `C_X(t)`, the characteristic function of a random variable `X`, following a Cauchy distribution: .. math:: C_X(t) = \exp(a \cdot i t - b |t|). .. code-block:: python >>> 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 .. method:: dist_cauchy.m_x(t) Returns ``NaN``, since the moment generating function does not exist. .. method:: dist_cauchy.k_x(t, k = 0) Returns ``NaN``, since the cumulant generating function does not exist. .. method:: dist_cauchy.moments(k) Returns ``NaN``, since moments do not exist. .. method:: dist_cauchy.cumulants(k) Returns ``NaN``, since cumulants do not exist.