Boost: Lognormal (Johnson \(S_L\)) distribution

The following functions return the pdf, cdf, qtf or boost class of the lognormal distribution with location \(a \in \mathbb{R}\), scale \(b > 0\), and the support interval \((0, +\infty)\).

See also Wikipedia [1250], MathWorld [900], BoostMath [67], Ehrhardt [309] (3.9.19).

Ctx.lognormal_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 lognormal distribution:

\[\text{pdf}(x) = \frac{1}{b x \sqrt{2\pi}} \exp \left(- \frac{(\log(x) - a)^2}{2b^2}\right).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("LognormalPdf(x, a, b): ", LognormalPdf(x, a, b))
>>> print ("dist_lognormal(a, b).pdf(x): ", dist_lognormal(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.lognormal_cdf(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 lognormal distribution:

\[\text{cdf}(x) = \frac{1}{2} \text{erfc} \left( -\frac{\log(x) - a}{b\sqrt{2}}\right).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("LognormalCdf(x, a, b): ", LognormalCdf(x, a, b))
>>> print ("dist_lognormal(a, b).cdf(x): ", dist_lognormal(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

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

where Ctx is Math53 or CtxBoost.

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

\[\text{qtf}(q) = \exp \left( a - b \sqrt{2} \cdot \text{erfc}^{-1}(2q) \right).\]

The following example shows both forms of the syntax:

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

class ctx.dist_lognormal(a, b)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

The lognormal distribution is a continuous probability distribution with location \(a \in \mathbb{R}\), scale \(b > 0\), and the support interval \((0, +\infty)\). See also Wikipedia [1250], MathWorld [900], BoostMath [67], Witkovský [1623], R (Statistical System) [551].

dist_lognormal.pdf(x)

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

\[\text{pdf}_X(x) = \frac{1}{b x \sqrt{2\pi}} \exp \left(- \frac{(\log(x) - a)^2}{2b^2}\right).\]

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

dist_lognormal.cdf(x)

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

\[\text{cdf}_X(x) = \frac{1}{2} \text{erfc} \left( -\frac{\log(x) - a}{b\sqrt{2}}\right).\]

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

dist_lognormal.sf(x)

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

\[\text{sf}_X(x) = \frac{1}{2} \text{erfc} \left( \frac{\log(x) - a}{b\sqrt{2}}\right).\]

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

dist_lognormal.qtf(q)

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

\[\text{qtf}_X(q) = \exp \left( a - b \sqrt{2} \cdot \text{erfc}^{-1}(2q) \right).\]

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

dist_lognormal.isf(q)

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

\[\text{isf}_X(q) = \exp \left( a + b \sqrt{2} \cdot \text{erfc}^{-1}(2q) \right).\]

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

dist_lognormal.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following an lognormal distribution. The characteristic function is defined for real values of \(t\), but is not defined for any complex value of \(t\) that has a negative imaginary part, and hence the characteristic function is not analytic at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. A closed-form formula for the characteristic function in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by

\[C_X(t) = \frac{\exp(-\frac{V^2+2V}{2\sigma^2})}{\sqrt{1+V}}, \quad \text{where } V = W(-it\sigma^2e^\mu).\]

where \(W\) is the Lambert \(W\) function. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence.

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

dist_lognormal.m_x(t)

The moment generating function does not exist.

dist_lognormal.k_x(t, k=0)

The cumulant generating function does not exist.

dist_lognormal.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an lognormal distribution. All moments of the log-normal distribution exist.

\[\mu'_{X}(r) = \exp(r \mu + r^2 \sigma^2 /2).\]

However, the log-normal distribution is not determined by its moments. This implies that it cannot have a defined moment generating function in a neighborhood of zero.

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

dist_lognormal.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following an lognormal distribution. The cumulants are calculated from the moments.

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