Noncentral Chi distribution

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

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

See also: Wikipedia [1308], Witkovský [1643].

dist_chi_nc.pdf(x)

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

\[\text{pdf}_X(x) = \frac{e^{-(x^2+\lambda^2)/2}x^k\lambda}{(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)\]

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

dist_chi_nc.cdf(x)

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

\[\text{cdf}_X(x) = 1 - Q_{\frac{k}{2}} \left( \lambda, x \right)\]

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

dist_chi_nc.sf(x)

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

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x).\]

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

dist_chi_nc.qtf(q)

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

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

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

dist_chi_nc.isf(q)

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

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

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

dist_chi_nc.c_x(t)

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

\[C_X(t) = ??\]

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

dist_chi_nc.m_x(t)

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

\[M_X(t) = ??\]

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

dist_chi_nc.k_x(t, k=0)

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

\[K_X(t) = ??\]

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

dist_chi_nc.moments(k)

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

\[\mu_{X}(k) = 2^{k/2} \Gamma(1+k/2) L_{k/2}^{((\nu-2)/2)}\left(-\tfrac{1}{2}\lambda^2\right)\]

\[L_n^{(\alpha)}(x) = \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+1)\Gamma(n+1)} {}_1F_1(-n;\alpha+1; x)\]

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

dist_chi_nc.cumulants(k)

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

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