.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| .. _rst_dist_laplace: Boost: Laplace distribution ------------------------------------------------------------------------------- The following functions return the pdf, cdf, qtf or boost class of the Laplace distribution with parameters `a \in \mathbb{R}` (location), `b > 0` (scale), and the support interval `(-\infty, +\infty)`. See also Wikipedia :cite:p:`WikipediaDis17`, MathWorld :cite:p:`WolframDis17`, BoostMath :cite:p:`BoostDis17`, :cite:t:`Ehrhardt2018` (3.9.15). |cr| .. _Ctx_LaplacePdf: .. method:: Ctx.laplace_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 Laplace distribution: .. math:: \text{pdf}(x) = \exp(- \vert x-a \vert /b)/(2b). 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 ("LaplacePdf(x, a, b): ", LaplacePdf(x, a, b)) >>> print ("dist_laplace(a, b).pdf(x): ", dist_laplace(a, b).pdf(x)) 6.3563523462564525615615615614561356E+00 |cr| .. _Ctx_LaplaceCdf: .. method:: Ctx.LaplaceCdf(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 Laplace distribution: .. math:: \text{cdf}(x) = \begin{cases} \frac{1}{2} - \frac{1}{2} \text{expm1}\left(- \frac{x-a}{b}\right) & x \geq a\\ \frac{1}{2} \text{exp}\left(- \frac{x-a}{b}\right) & x>> from mpfebnet import * >>> a = 0; b = 1; t = 0.3; x = 0.6; >>> print ("LaplaceCdf(x, a, b): ", LaplaceCdf(x, a, b)) >>> print ("dist_laplace(a, b).cdf(x): ", dist_laplace(a, b).cdf(x)) 6.3563523462564525615615615614561356E+00 |cr| .. _Ctx_LaplaceQtf: .. method:: Ctx.LaplaceQtf(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 Laplace distribution: .. math:: \text{qtf}(q) = \begin{cases} a+b \: \log(2q), & q \le 0.5,\\ a-b \: \log(2(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 ("LaplaceQtf(q, a, b): ", LaplaceQtf(q, a, b)) >>> print ("dist_laplace(a, b).qtf(q): ", dist_laplace(a, b).qtf(q)) 6.3563523462564525615615615614561356E+00 |cr| .. py:class:: ctx.dist_laplace(a, b) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. The Laplace 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:`WikipediaDis17`, MathWorld :cite:p:`WolframDis17`, BoostMath :cite:p:`BoostDis17`, :cite:t:`CharfunDis17`. |cr| .. method:: dist_laplace.pdf(x) Returns `\text{pdf}_X(x)`, the probability density function (pdf) of a random variable `X`, following an Laplace distribution: .. math:: \text{pdf}_X(x) = \exp(- \vert x-a \vert /b)/(2b). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("pdf: ", laplace(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_laplace.cdf(x) Returns `\text{cdf}_X(x)`, the cumulative distribution function (cdf) of a random variable `X`, following an Laplace distribution: .. math:: \text{cdf}_X(x) = \begin{cases} \frac{1}{2} - \frac{1}{2} \text{expm1}\left(- \frac{x-a}{b}\right) & x \geq a\\ \frac{1}{2} \text{exp}\left(- \frac{x-a}{b}\right) & x>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("cdf: ", laplace(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_laplace.sf(x) Returns `\text{sf}_X(x)`, the survival function (sf) of a random variable `X`, following an Laplace distribution: .. math:: \text{sf}_X(x) = \begin{cases} \frac{1}{2} + \frac{1}{2} \text{expm1}\left(- \frac{x-a}{b}\right) & x \geq a\\ 1 - \frac{1}{2} \text{exp}\left(- \frac{x-a}{b}\right) & x>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print (" sf: ", laplace(mu, sigma).pdf(x)) sf: 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_laplace.qtf(q) Returns `\text{qtf}_X(x)`, the quantile function (qtf) of a random variable `X`, following an Laplace distribution: .. math:: \text{qtf}_X(q) = \begin{cases} a+b \: \log(2q), & q \le 0.5,\\ a-b \: \log(2(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: ", laplace(mu, sigma).qtf(q)) qtf: 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_laplace.isf(q) Returns `\text{isf}_X(q)`, the inverse survival function (isf) of a random variable `X`, following an Laplace distribution: .. math:: \text{isf}_X(q) = \begin{cases} a-b \: \log(2q), & q \le 0.5,\\ a+b \: \log(2(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: ", laplace(mu, sigma).isf(q)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_laplace.c_x(t) Returns `C_X(t)`, the characteristic function of a random variable `X`, following an Laplace distribution: .. math:: C_X(t) = \frac{e^{iat}}{1-b^2 t^2}. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", laplace(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_laplace.m_x(t) Returns `M_X(t)`, the moment generating function of a random variable `X`, following an Laplace distribution: .. math:: M_X(t) = \frac{e^{at}}{1-b^2 t^2}. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("m_x: ", laplace(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_laplace.k_x(t, k = 0) Returns `K_X(t)`, the cumulant generating function of a random variable `X`, following an Laplace distribution: .. math:: K_X(t) = at - \log(1-b^2 t^2. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; k = 6; >>> print ("k_x: ", laplace(mu, sigma).k_x(t, k)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_laplace.moments(k) Returns the first `j` raw moments, `\mu_j, j = 1 \ldots k`, of a random variable `X`, following an Laplace distribution. The moments are calculated from the cumulants. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; k = 6; >>> print ("saddlepoint: ", laplace(mu, sigma).moments(k)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_laplace.cumulants(k) Returns the first `j` cumulants, `\kappa_j, j = 1 \ldots k`, of a random variable `X`, following an Laplace distribution. The cumulants are calculated from the moments. .. math:: \kappa_{X}(r) =\begin{cases} a & r = 1\\ 0 & r = 2k+1; k=0,1,\ldots\\ b^r 2(r-1)! & r = 2k; k=0,1,\ldots \end{cases} .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; k = 6; >>> print ("saddlepoint: ", laplace(mu, sigma).cumulants(k)) 6.3563523462564525615615615614561356E+00