Half-normal distribution

class ctx.dist_half_normal(sigma)

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

The half-normal distribution is a continuous probability distribution with standard deviation \(\sigma > 0\), and the support interval \((0, +\infty)\). See also Wikipedia [1269], MathWorld [896], Witkovský [1638], .

dist_half_normal.pdf(x)

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

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

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

dist_half_normal.cdf(x)

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

\[\text{cdf}_X(x) = \text{erf}\left(\frac{x}{\sigma \sqrt{2}} \right).\]

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

dist_half_normal.sf(x)

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

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

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

dist_half_normal.qtf(q)

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

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

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

dist_half_normal.isf(q)

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

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

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

dist_half_normal.c_x(t)

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

\[C_X(t) = M\left( \frac{3}{2}, \frac{1}{2} \frac{-t^2}{2} \right) + \frac{2 \sqrt{2} \, i t }{\sqrt{\pi}} M\left(2, \frac{3}{2}, \frac{-t^2}{2} \right).\]

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

dist_half_normal.m_x(t)

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

\[M_X(t) = M\left( \frac{3}{2}, \frac{1}{2} \frac{t^2}{2} \right) + \frac{2 \sqrt{2} \, t }{\sqrt{\pi}} M\left( 2, \frac{3}{2}, \frac{t^2}{2} \right).\]

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

dist_half_normal.k_x(t, k=0)

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

\[K_X(t) = \log \left[ M\left( \frac{3}{2}, \frac{1}{2} \frac{t^2}{2} \right) + \frac{2 \sqrt{2} \, t }{\sqrt{\pi}} M\left(2, \frac{3}{2}, \frac{t^2}{2} \right) \right].\]

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

dist_half_normal.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an half-normal distribution (Wikipedia). The raw moments are calculated from the central moments.

\[\begin{split}\mu_{X}(r) = \begin{cases} \sqrt{\frac{2}{\pi}} k! \alpha^{2k-1} & \text{for } n=2k-1,\\ (n+1)!! \alpha^n & \text{for } n \text{ even}, \end{cases}\end{split}\]

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

dist_half_normal.cumulants(k)

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

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