Normal range distribution

class ctx.dist_normal_range(k)

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

The normal range distribution is a continuous probability distribution with \(k \ge 2\) groups, and the support interval \((0, +\infty)\). See also Wikipedia [1310], Harter [381], R (Statistical System) [561], and dist_normal_range().

dist_normal_range.pdf(x)

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

\[\text{pdf}_X(x) = k(k-1) \int_{-\infty}^\infty \left( \Phi(y) - \Phi(y-x) \right)^{k-2} \phi(y) \phi(y-x) \mathrm{d} y.\]

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

dist_normal_range.cdf(x)

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

\[\text{cdf}_X(x) = k \int_{-\infty}^\infty \phi(y) \left( \Phi(y) - \Phi(y-x) \right)^{k-1} \mathrm{d} y.\]

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

dist_normal_range.sf(x)

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

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = k \int_{-\infty}^\infty \phi(y) \left( L_1^k - :cite:t:`L_1 - L_2]^{k-1} \right) \mathrm{d} y, \quad \text{where}\]

\[L_1 = \Phi(y), \quad L_2 = \Phi(y-x), \quad L_2 \rightarrow 0 \text{ for } x \rightarrow \infty\]

\[L_1^k - (L_1 - L_2)^k = L_1^k \left( 1 - \left( 1 - \frac{L_2}{L_1} \right) ^k \right)\]

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

dist_normal_range.qtf(q)

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

There is no known explicit form for the quantile function \(\text{qtf}_X(x)\): It is computed using Newton iterations with starting values from a central \(F\) approximation.

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

dist_normal_range.isf(q)

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

There is no known explicit form for the quantile function \(\text{isf}_X(x)\): It is computed using Newton iterations with starting values from a central \(F\) approximation.

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

dist_normal_range.c_x(t)

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

\[C_X(t) = \int_{0}^{\infty} e^{i tx} \text{pdf}_X(x) \mathrm{d} x\]

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

dist_normal_range.m_x(t)

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

\[M_X(t) = \int_{0}^{\infty} e^{tx} \text{pdf}_X(x) \mathrm{d} x\]

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

dist_normal_range.k_x(t, k=0)

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

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

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

dist_normal_range.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a central normal range distribution. The rth moments only exists for \(n_2 > 2r\).

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

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

dist_normal_range.cumulants(k)

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

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

Additional information

\[Q(x) = k \int_{-\infty}^\infty \phi(y) \left( \Phi(y) - \Phi(y-x) \right)^{k-1} \mathrm{d} y\]

\[P(x) = 1 - Q(x) = k \int_{-\infty}^\infty \phi(y) \left( L_1^k - :cite:t:`L_1 - L_2]^{k-1} \right) \mathrm{d} y, \quad \text{where}\]

\[L_1 = \Phi(y), \quad L_2 = \Phi(y-x), \quad L_2 \rightarrow 0 \text{ for } x \rightarrow \infty\]

\[L_1^k - (L_1 - L_2)^k = L_1^k \left( 1 - \left( 1 - \frac{L_2}{L_1} \right) ^k \right)\]