Truncated normal distribution

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

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

See also: Wikipedia [1311], Orjebin [475], Burkardt [171],

John Burkardt: The Truncated Normal Distribution (has closed form qtf)

dist_truncated_normal.pdf(x)

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

\[\text{pdf}_X(x) = f(x;\mu ,\sigma ,a,b)={\frac {1}{\sigma }}\, \frac {\phi ({\frac {x-\mu }{\sigma }})}{\Phi \left({\frac {b-\mu }{\sigma }}\right)-\Phi \left({\frac {a-\mu }{\sigma }}\right)}\]

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

dist_truncated_normal.cdf(x)

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

\[\text{cdf}_X(x) = \frac{ \Phi(\xi) - \Phi(\alpha) }{\Phi(\beta) - \Phi(\alpha)}\]

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

dist_truncated_normal.sf(x)

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

\[\text{sf}_X(x) = 1 - \frac{ \Phi(\xi) - \Phi(\alpha) }{\Phi(\beta) - \Phi(\alpha)}\]

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

dist_truncated_normal.qtf(q)

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

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

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

dist_truncated_normal.isf(q)

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

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

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

dist_truncated_normal.c_x(t)

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

\[C_X(t) = e^{\mu it - \sigma^2 t^2 /2} \cdot \frac{ \Phi(\beta - \sigma it) - \Phi(\alpha- \sigma it) }{\Phi(\beta) - \Phi(\alpha)}\]

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

dist_truncated_normal.m_x(t)

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

\[M_X(t) = e^{\mu t + \sigma^2 t^2 /2} \cdot \frac{ \Phi(\beta - \sigma t) - \Phi(\alpha- \sigma t) }{\Phi(\beta) - \Phi(\alpha)}\]

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

dist_truncated_normal.k_x(t, k=0)

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

\[K_X(t) = \log (M_X(t))\]

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

dist_truncated_normal.moments(k)

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

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

A recursive formula for \(\mu_k\) (based on \(\mu_{-1}=0\) and \(\mu_0=1\)) is:

\[\mu'_X(r) = m_k = (k-1) \sigma^2 m_{k-2} + \mu m_{k-1} - \sigma \frac{b^{k-1} \phi \left({\frac {b-\mu }{\sigma }} \right)- a^{k-1} \phi \left({\frac {a-\mu }{\sigma }} \right)}{\Phi \left({\frac {b-\mu }{\sigma }} \right)-\Phi \left({\frac {a-\mu }{\sigma }} \right)}\]

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

dist_truncated_normal.cumulants(k)

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

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