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 [1249], MathWorld [899], BoostMath [66], Ehrhardt [309] (3.9.15).

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 (Pdf) of the Laplace distribution:

\[\text{pdf}(x) = \exp(- \vert x-a \vert /b)/(2b).\]

The following example shows both forms of the syntax:

>>> 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

Ctx.LaplaceCdf(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 Laplace distribution:

\[\begin{split}\text{cdf}(x) = \begin{cases} \frac{1}{2} - \frac{1}{2} \text{expm1}\left(- \frac{x-a}{b}\right) & x \geq a\\ \frac{1}{2} \text{exp}\left(- \frac{x-a}{b}\right) & x<a. \end{cases}\end{split}\]

The following example shows both forms of the syntax:

>>> 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

Ctx.LaplaceQtf(q, a=0, b=1)

where Ctx is Math53 or CtxBoost.

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

\[\begin{split}\text{qtf}(q) = \begin{cases} a+b \: \log(2q), & q \le 0.5,\\ a-b \: \log(2(1-q)) & q>0.5. \end{cases}\end{split}\]

The following example shows both forms of the syntax:

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

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 [1249], MathWorld [899], BoostMath [66], Witkovský [1622].

dist_laplace.pdf(x)

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

\[\text{pdf}_X(x) = \exp(- \vert x-a \vert /b)/(2b).\]

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

dist_laplace.cdf(x)

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

\[\begin{split}\text{cdf}_X(x) = \begin{cases} \frac{1}{2} - \frac{1}{2} \text{expm1}\left(- \frac{x-a}{b}\right) & x \geq a\\ \frac{1}{2} \text{exp}\left(- \frac{x-a}{b}\right) & x<a. \end{cases}\end{split}\]

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

dist_laplace.sf(x)

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

\[\begin{split}\text{sf}_X(x) = \begin{cases} \frac{1}{2} + \frac{1}{2} \text{expm1}\left(- \frac{x-a}{b}\right) & x \geq a\\ 1 - \frac{1}{2} \text{exp}\left(- \frac{x-a}{b}\right) & x<a. \end{cases}\end{split}\]

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

dist_laplace.qtf(q)

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

\[\begin{split}\text{qtf}_X(q) = \begin{cases} a+b \: \log(2q), & q \le 0.5,\\ a-b \: \log(2(1-q)) & q>0.5. \end{cases}\end{split}\]

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

dist_laplace.isf(q)

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

\[\begin{split}\text{isf}_X(q) = \begin{cases} a-b \: \log(2q), & q \le 0.5,\\ a+b \: \log(2(1-q)) & q>0.5. \end{cases}\end{split}\]

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

dist_laplace.c_x(t)

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

\[C_X(t) = \frac{e^{iat}}{1-b^2 t^2}.\]

>>> 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

dist_laplace.m_x(t)

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

\[M_X(t) = \frac{e^{at}}{1-b^2 t^2}.\]

>>> 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

dist_laplace.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following an Laplace distribution:

\[K_X(t) = at - \log(1-b^2 t^2.\]

>>> 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

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.

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

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.

\[\begin{split}\kappa_{X}(r) =\begin{cases} a & r = 1\\ 0 & r = 2k+1; k=0,1,\ldots\\ b^r 2(r-1)! & r = 2k; k=0,1,\ldots \end{cases}\end{split}\]

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