Folded normal distribution

class ctx.dist_folded_normal(n1, n2, lambda, **kwargs)

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

These functions return PDF, CDF, and ICDF of the folded normal distribution with location \(a\), scale \(b > 0\), and the support interval \((-\infty,+\infty)\) :

See also: Wikipedia [1295], Tsagris et al. [859], Reig et al. [500].

dist_folded_normal.pdf(x)

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

\[\text{pdf}_X(x) = f_{Y}(x;\mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}+{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\,e^{-{\frac {(x+\mu )^{2}}{2\sigma ^{2}}}}\]

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

dist_folded_normal.cdf(x)

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

\[\text{cdf}_X(x) = F_{Y}(x;\mu ,\sigma ^{2}) = {\frac {1}{2}}\left[{\mbox{erf}}\left({\frac {x+\mu }{\sqrt {2\sigma ^{2}}}}\right)+{\mbox{erf}}\left({\frac {x-\mu }{\sqrt {2\sigma ^{2}}}}\right)\right]\]

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

dist_folded_normal.sf(x)

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

\[\text{sf}_X(x) = 1-F_{Y}(x;\mu ,\sigma ^{2}) = 1-{\frac {1}{2}}\left[{\mbox{erf}}\left({\frac {x+\mu }{\sqrt {2\sigma ^{2}}}}\right)+{\mbox{erf}}\left({\frac {x-\mu }{\sqrt {2\sigma ^{2}}}}\right)\right]\]

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

dist_folded_normal.qtf(q)

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

\[\text{qtf}_X(q) = ??\]

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

dist_folded_normal.isf(q)

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

\[\text{isf}_X(q) = ??\]

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

dist_folded_normal.c_x(t)

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

\[C_X(t) = e^{{\frac {-\sigma ^{2}t^{2}}{2}}+i\mu t}\Phi \left({\frac {\mu }{\sigma }}+i\sigma t\right)+e^{-{\frac {\sigma ^{2}t^{2}}{2}}-i\mu t}\Phi \left(-{\frac {\mu }{\sigma }}+i\sigma t\right).\]

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

dist_folded_normal.m_x(t)

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

\[M_X(t) = e^{{\frac {\sigma ^{2}t^{2}}{2}}+\mu t}\Phi \left({\frac {\mu }{\sigma }}+\sigma t\right)+e^{{\frac {\sigma ^{2}t^{2}}{2}}-\mu t}\Phi \left(-{\frac {\mu }{\sigma }}+\sigma t\right).\]

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

dist_folded_normal.k_x(t, k=0)

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

\[K_X(t) = \left({\frac {\sigma ^{2}t^{2}}{2}}+\mu t\right)+\log {\left\lbrace 1-\Phi \left(-{\frac {\mu }{\sigma }}-\sigma t\right)+e^{{\frac {\sigma ^{2}t^{2}}{2}}-\mu t}\left[1-\Phi \left({\frac {\mu }{\sigma }}-\sigma t\right)\right]\right\rbrace }.\]

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

dist_folded_normal.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an folded normal distribution. The moments are calculated from their definition:

\[\mu'_X(r) = E(X^r) = \int_{0}^{1} x^r \text{pdf}_X(x) \mathrm{d} x\]

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

dist_folded_normal.cumulants(k)

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

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