!!!Boost: Chi Distribution

The following functions return the pdf, cdf, qtf or boost class of the chi distribution with \(n > 0\) degrees of freedom and the support interval \((0,+\infty)\).

See also Wikipedia [1267], MathWorld [891], Ehrhardt [309] (3.9.5)

Ctx.chi_pdf(x, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the chi distribution:

\[\text{pdf}(x) = 2x \times f_{\chi^2}\left(x^2, n\right).\]

Here \(f_{\chi^2}(x,n)\) denotes the probability density function of a random variable following an chi-squared distribution with \(n\) degress of freedom.

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("ChiPdf(x, a, b): ", ChiPdf(x, a, b))
>>> print ("dist_chi(a, b).pdf(x): ", dist_chi(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.chi_cdf(x, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the chi distribution:

\[\text{cdf}(x) = P(n/2, x^2/2).\]

Here \(P(\cdot)\) denotes the lower regularized incomplete gamma function (RealGammaP).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("ChiCdf(x, a, b): ", ChiCdf(x, a, b))
>>> print ("dist_chi(a, b).cdf(x): ", dist_chi(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.chi_qtf(q, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the chi distribution:

\[\text{qtf}(q) = \sqrt{2 P^{-1}(n/2, q)}.\]

Here \(P^{-1}(\cdot)\) denotes the inverse of the lower regularized incomplete gamma function (RealGammaPInv).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("ChiQtf(q, a, b): ", ChiQtf(q, a, b))
>>> print ("dist_chi(a, b).qtf(q): ", dist_chi(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_chi(n)

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

The chi distribution is a continuous probability distribution with \(n > 0\) degrees of freedom and the support interval \((0,+\infty)\). See also Wikipedia [1267], MathWorld [891], Witkovský [1635].

dist_chi.pdf(x)

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

\[\text{pdf}_X(x) = 2x \times f_{\chi^2}\left(x^2, n\right).\]

Here \(f_{\chi^2}(x,n)\) denotes the probability density function of a random variable following an chi-squared distribution with \(n\) degress of freedom.

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

dist_chi.cdf(x)

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

\[\text{cdf}_X(x) = P(n/2, x^2/2).\]

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

dist_chi.sf(x)

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

\[\text{sf}_X(x) = Q(n/2, x^2/2).\]

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

dist_chi.qtf(q)

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

\[\text{qtf}_X(q) = \sqrt{2 P^{-1}(n/2, q)}.\]

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

dist_chi.isf(q)

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

\[\text{isf}_X(q) = \sqrt{2 Q^{-1}(n/2, q)}.\]

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

dist_chi.c_x(t)

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

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

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

dist_chi.m_x(t)

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

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

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

dist_chi.k_x(t, k=0)

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

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

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

dist_chi.moments(k)

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

\[\mu_{X}(j) = 2^{2j} \frac{\Gamma\left( (k+j)/2 \right)}{\Gamma(k/2)} .\]

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

dist_chi.cumulants(k)

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

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