Boost: Normal (Johnson \(S_N\)) distribution#

The following functions return the pdf, cdf, qtf or boost class of the normal distribution with mean \(\mu \in \mathbb{R}\), standard deviation \(\sigma > 0\), and the support interval \((-\infty, +\infty)\).

See also Wikipedia [1258], MathWorld [906], BoostMath [73], Ehrhardt [309] (3.9.24).

Ctx.normal_pdf(x, mu=0, sigma=1)#

where Ctx is Math53 or CtxBoost.

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

\[\text{pdf}(x) = \phi(x) = {\frac {1}{\sqrt{ 2 \pi \sigma^2 }}} e ^{- {\frac {(x-\mu)^2 }{2\sigma^2 }}}.\]

The following example shows both forms of the syntax:

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

Ctx.normal_cdf(x, mu=0, sigma=1)#

where Ctx is Math53 or CtxBoost.

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

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

The following example shows both forms of the syntax:

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

Ctx.normal_qtf(q, mu=0, sigma=1)#

where Ctx is Math53 or CtxBoost.

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

\[\text{qtf}(q) = \mu - \sigma \sqrt{2} \cdot \text{erfc}^{-1}(2q).\]

The following example shows both forms of the syntax:

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

class ctx.dist_normal(mu, sigma)#

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

The normal distribution is a continuous probability distribution with mean \(\mu \in \mathbb{R}\), standard deviation \(\sigma > 0\), and the support interval \((-\infty, +\infty)\). See also Wikipedia [1258], MathWorld [906], BoostMath [73], R (Statistical System) [552], Mpmath [569], Mpmath [565].

dist_normal.pdf(x)#

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

\[\text{pdf}_X(x) = \phi(x) = {\frac {1}{\sqrt{ 2 \pi \sigma^2 }}} e ^{- {\frac {(x-\mu)^2 }{2\sigma^2 }}}.\]

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

dist_normal.cdf(x)#

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

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

Here \(\text{erf}(\cdot)\) and \(\text{erf}^{-1}(\cdot)\) are the error function and its functional inverse, respectively.

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

dist_normal.sf(x)#

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

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

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

dist_normal.qtf(q)#

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

\[\text{qtf}_X(q) = \mu - \sigma \sqrt{2} \cdot \text{erfc}^{-1}(2q).\]

Here \(\text{erf}(\cdot)\) and \(\text{erf}^{-1}(\cdot)\) are the error function and its functional inverse, respectively.

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

dist_normal.isf(q)#

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

\[\text{isf}_X(q) = \mu + \sigma \sqrt{2} \cdot \text{erfc}^{-1}(2q).\]

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

dist_normal.c_x(t)#

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

\[C_X(t) = \exp \left( i \mu t - \tfrac{1}{2} \sigma^2 t^2 \right).\]

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

dist_normal.m_x(t)#

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

\[M_X(t) = \exp \left( \mu t + \tfrac{1}{2} \sigma^2 t^2 \right).\]

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

dist_normal.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 normal distribution:

\[K_X(t) = \mu t + \tfrac{1}{2} \sigma^2 t^2 ,\]

\[K_X^{(1)}(t) = \mu + \sigma^2 t, \quad K_X^{(2)}(t) = \sigma^2, \quad K_X^{(j)}(t) = 0, \quad j > 2.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", normal(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_normal.moments(k)#

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a normal distribution: the moments are calculated from the cumulants.

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

dist_normal.cumulants(k)#

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

\[\kappa_{1} = \mu, \quad \kappa_{2} = \sigma^2, \quad \kappa_{j} = 0, \quad j > 2.\]

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

Boost: Normal (Johnson S_N) distribution

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Boost: Normal (Johnson \(S_N\)) distribution#