Boost: Inverse Gamma (Pearson Type V) distribution

Returns the pdf, cdf, qtf or boost class of a random variable \(X\), following an inverse gamma distribution with shape \(a > 0\), scale \(b > 0\), and the support interval \((0,+\infty)\).

See also Wikipedia [1248], BoostMath [64], Witkovský [1610], Ehrhardt [309] (3.9.12).

Ctx.invgamma_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 inverse gamma distribution:

\[\text{pdf}(x) = \left(\frac{b}{x}\right)^a \frac{e^{-\frac{b}{x}}}{x\Gamma(a)}.\]

The following example shows both forms of the syntax:

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

Ctx.invgamma_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 inverse gamma distribution:

\[\text{cdf}(x) = Q \left(a, -\frac{b}{x} \right).\]

Here \(Q(\cdot)\) denotes the upper regularized incomplete gamma function (RealGammaQ).

The following example shows both forms of the syntax:

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

Ctx.invgamma_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 inverse gamma distribution:

\[\text{qtf}(q) = \frac{b}{Q^{-1}(a, q)}.\]

Here \(Q^{-1}(\cdot)\) denotes the inverse of the upper regularized incomplete gamma function (RealGammaQInv).

The following example shows both forms of the syntax:

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

class ctx.dist_invgamma(a, b)

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

The inverse gamma distribution is a continuous probability distribution with shape \(a > 0\), scale \(b > 0\), and the support interval \((0,+\infty)\). See also Wikipedia [1248], MathWorld [251], BoostMath [64], Witkovský [1620], Witkovský [1610].

dist_invgamma.pdf(x)

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

\[\text{pdf}_X(x) = \left(\frac{b}{x}\right)^a \frac{e^{-\frac{b}{x}}}{x\Gamma(a)} .\]

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

dist_invgamma.cdf(x)

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

\[\text{cdf}_X(x) = Q \left(a, -\frac{b}{x} \right).\]

where \(Q(\cdot)\) and \(Q^{-1}(\cdot)\) denote the regularized gamma function and its functional inverse.

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

dist_invgamma.sf(x)

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

\[\text{sf}_X(x) = P \left(a, -\frac{b}{x} \right).\]

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

dist_invgamma.qtf(q)

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

\[\text{qtf}_X(q) = \frac{b}{Q^{-1}(a, q)}.\]

where \(Q(\cdot)\) and \(Q^{-1}(\cdot)\) denote the regularized gamma function and its functional inverse.

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

dist_invgamma.isf(q)

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

\[\text{isf}_X(q) = \frac{b}{P^{-1}(a, q)}.\]

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

dist_invgamma.c_x(t)

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

\[C_X(t) = \frac{2(-i \beta t)^{\alpha /2}}{\Gamma(\alpha)} K_{\alpha}(\sqrt{-4 i \beta t}),\]

where \(K_n(\cdot)\) denotes the modified Bessel function of the second kind.

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

dist_invgamma.m_x(t)

The moment generating function does not exist.

dist_invgamma.k_x(t, k=0)

The cumulant generating function does not exist.

dist_invgamma.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an inverse gamma distribution. The rth moments only exists for \(\alpha > r\).

\[\mu'_X(r) = \frac{\beta^n}{(\alpha - 1) \cdots (\alpha - n)}\]

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

dist_invgamma.cumulants(k)

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

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