.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| .. _rst_dist_wald: Boost: Wald (or Inverse Gaussian) distribution =============================================================================== The following functions return the pdf, cdf, qtf or boost class of the Wald distribution with mean `\mu>0`, scale `b > 0`, and the support interval `(0, +\infty)`. See also Wikipedia :cite:p:`WikipediaDis16`, MathWorld :cite:p:`WolframDis16`, BoostMath :cite:p:`BoostDis16`, :cite:t:`Ehrhardt2018` (3.9.32). |cr| .. _Ctx_WaldPdf: .. method:: Ctx.wald_pdf(x, mu, b) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{pdf}(x)`, the value of the probability density function (:ref:`Pdf `) of the Wald distribution: .. math:: \text{pdf}(x) = \sqrt{\frac{b}{2\pi x^3}} \exp \left( \frac{-b(x-\mu)^2}{2\mu^2 x} \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 ("WaldPdf(x, a, b): ", WaldPdf(x, a, b)) >>> print ("dist_wald(a, b).pdf(x): ", dist_wald(a, b).pdf(x)) 6.3563523462564525615615615614561356E+00 |cr| .. _Ctx_WaldCdf: .. method:: Ctx.wald_cdf(x, mu, b) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{cdf}(x)`, the value of the cumulative distribution function (:ref:`Cdf `) of the Wald distribution: .. math:: \text{cdf}(x) = \Phi\left(\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}-1\right)\right) + \exp \left( \frac{2b}{\mu} \right) \Phi\left(-\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}+1\right)\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 ("WaldCdf(x, a, b): ", WaldCdf(x, a, b)) >>> print ("dist_wald(a, b).cdf(x): ", dist_wald(a, b).cdf(x)) 6.3563523462564525615615615614561356E+00 |cr| .. _Ctx_WaldQtf: .. method:: Ctx.wald_qtf(q, mu, b) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{qtf}(q)`, the value of the quantile function (:ref:`Qtf `) of the Wald distribution: There is no known closed form for `\text{qtf}(q)`: it is computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision). 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 ("WaldQtf(q, a, b): ", WaldQtf(q, a, b)) >>> print ("dist_wald(a, b).qtf(q): ", dist_wald(a, b).qtf(q)) 6.3563523462564525615615615614561356E+00 |cr| .. py:class:: ctx.dist_wald(mu, b) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. The Wald (or inverse Gaussian) distribution is a continuous probability distribution with mean `\mu>0`, scale `b > 0`, and the support interval `(0, +\infty)`. See also Wikipedia :cite:p:`WikipediaDis16`, MathWorld :cite:p:`WolframDis16`, BoostMath :cite:p:`BoostDis16`, :cite:t:`CharfunDis16`. |cr| .. method:: dist_wald.pdf(x) Returns `\text{pdf}_X(x)`, the probability density function (pdf) of a random variable `X`, following a Wald distribution: .. math:: \text{pdf}_X(x) = \sqrt{\frac{b}{2\pi x^3}} \exp \left( \frac{-b(x-\mu)^2}{2\mu^2 x} \right). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("pdf: ", wald(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_wald.cdf(x) Returns `\text{cdf}_X(x)`, the cumulative distribution function (cdf) of a random variable `X`, following a Wald distribution: .. math:: \text{cdf}_X(x) = \Phi\left(\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}-1\right)\right) + \exp \left( \frac{2b}{\mu} \right) \Phi\left(-\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}+1\right)\right). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("cdf: ", wald(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_wald.sf(x) Returns `\text{sf}_X(x)`, the survival function (sf) of a random variable `X`, following a Wald distribution: .. math:: \text{sf}_X(x) = \Phi\left(-\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}-1\right)\right) - \exp \left( \frac{2b}{\mu} \right) \Phi\left(-\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}+1\right)\right). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print (" sf: ", wald(mu, sigma).pdf(x)) sf: 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_wald.qtf(q) Returns `\text{qtf}_X(x)`, the quantile function (qtf) of a random variable `X`, following a Wald distribution: There is no known closed form for `\text{qtf}_X(q)` or `\text{isf}_X(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("qtf: ", wald(mu, sigma).qtf(q)) qtf: 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wald.isf(q) Returns `\text{isf}_X(q)`, the inverse survival function (isf) of a random variable `X`, following a Wald distribution: There is no known closed form for `\text{qtf}_X(q)` or `\text{isf}_X(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("isf: ", wald(mu, sigma).isf(q)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wald.c_x(t) Returns `C_X(t)`, the characteristic function of a random variable `X`, following a Wald distribution: .. math:: C_X(t) = \exp \left[ \frac{\lambda}{\mu} \left( 1 - \sqrt{1 - \frac{2 \mu^2 it}{\lambda}} \right) \right] . .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", wald(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wald.m_x(t) Returns `M_X(t)`, the moment generating function of a random variable `X`, following a Wald distribution: .. math:: M_X(t) = \exp \left[ \frac{\lambda}{\mu} \left( 1 - \sqrt{1 - \frac{2 \mu^2 t}{\lambda}} \right) \right]. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("m_x: ", wald(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wald.k_x(t, k = 0) Returns `K_X(t)`, the cumulant generating function, and its `j^{\text{th}}` derivatives, `K_X^{(j)}(t), j = 1 \ldots k`, of a random variable `X`, following a Wald distribution: .. math:: K_X(t) = \frac{\lambda}{\mu} \left( 1 - \sqrt{1 - \frac{2 \mu^2 t}{\lambda}} \right). .. math:: K_X^{(j)}(t) = tbd . .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; k = 6; >>> print ("c_x: ", wald(mu, sigma).k_x(t, k)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wald.moments(k) Returns the first `j` raw moments, `\mu_j, j = 1 \ldots k`, of a random variable `X`, following a Wald 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: ", wald(mu, sigma).moments(k)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wald.cumulants(k) Returns the first `j` cumulants, `\kappa_j, j = 1 \ldots k`, of a random variable `X`, following a Wald distribution. .. math:: \kappa_X(n+1) =\kappa_{X, n+1} = \frac{(2n)!}{2^n n! \mu^{2n+1} \lambda^n} .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; k = 6; >>> print ("saddlepoint: ", wald(mu, sigma).cumulants(k)) 6.3563523462564525615615615614561356E+00