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 [1247], MathWorld [898], BoostMath [65], Ehrhardt [309] (3.9.32).

Ctx.wald_pdf(x, mu, b)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the Wald distribution:

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

The following example shows both forms of the syntax:

>>> 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

Ctx.wald_cdf(x, mu, b)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the Wald distribution:

\[\text{cdf}(x) = \Phi\left(\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}-1\right)\right) + \exp \left( \frac{2b}{\mu} \right) \Phi\left(-\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}+1\right)\right).\]

The following example shows both forms of the syntax:

>>> 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

Ctx.wald_qtf(q, mu, b)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (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:

>>> 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

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 [1247], MathWorld [898], BoostMath [65], Witkovský [1621].

dist_wald.pdf(x)#

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

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

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

dist_wald.cdf(x)#

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

\[\text{cdf}_X(x) = \Phi\left(\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}-1\right)\right) + \exp \left( \frac{2b}{\mu} \right) \Phi\left(-\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}+1\right)\right).\]

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

dist_wald.sf(x)#

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

\[\text{sf}_X(x) = \Phi\left(-\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}-1\right)\right) - \exp \left( \frac{2b}{\mu} \right) \Phi\left(-\sqrt{\frac{b}{x}} \left(\frac{x}{\mu}+1\right)\right).\]

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

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).

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

dist_wald.isf(q)#

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

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

dist_wald.c_x(t)#

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

\[C_X(t) = \exp \left[ \frac{\lambda}{\mu} \left( 1 - \sqrt{1 - \frac{2 \mu^2 it}{\lambda}} \right) \right] .\]

>>> 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

dist_wald.m_x(t)#

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

\[M_X(t) = \exp \left[ \frac{\lambda}{\mu} \left( 1 - \sqrt{1 - \frac{2 \mu^2 t}{\lambda}} \right) \right].\]

>>> 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

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:

\[K_X(t) = \frac{\lambda}{\mu} \left( 1 - \sqrt{1 - \frac{2 \mu^2 t}{\lambda}} \right).\]

\[K_X^{(j)}(t) = tbd .\]

>>> 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

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.

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

dist_wald.cumulants(k)#

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a Wald distribution.

\[\kappa_X(n+1) =\kappa_{X, n+1} = \frac{(2n)!}{2^n n! \mu^{2n+1} \lambda^n}\]

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

Boost: Wald (or Inverse Gaussian) distribution

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Boost: Wald (or Inverse Gaussian) distribution#