Beta-prime (Pearson Type VI) distribution

class ctx.dist_beta_prime(a, b)

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

The beta-prime distribution is a continuous probability distribution with parameters \(a > 0\) and \(b > 0\), and the support interval \((0, +\infty)\).

See also Wikipedia [1288], Becker [32].

Pearson Type I: Beta

Pearson Type II: Symmetric Beta

Pearson Type III: Gamma

Pearson Type IV: Extra

Pearson Type V: Inverse Gamma

Pearson Type VI: Beta Prime

Pearson Type VII: Student’s t

dist_beta_prime.pdf(x)

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

\[\text{pdf}_X(x) = f(x)=\frac {x^{{\alpha -1}}(1+x)^{{-\alpha -\beta }}}{B(\alpha ,\beta )}\]

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

dist_beta_prime.cdf(x)

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

\[\text{cdf}_X(x) = I_{\tfrac{x}{1+x}}(a,b)\]

Here \(\text{ibeta}(\cdot)\) denotes the real normalised incomplete beta function, and \(\text{ibetac}(\cdot)\) denotes the real normalised complementary incomplete beta function.

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

dist_beta_prime.sf(x)

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

\[\text{sf}_X(x) = 1 - I_{\tfrac{x}{1+x}}(a,b)\]

Here \(\text{ibeta}(\cdot)\) denotes the real normalised incomplete beta function, and \(\text{ibetac}(\cdot)\) denotes the real normalised complementary incomplete beta function.

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

dist_beta_prime.qtf(q)

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

\[\text{qtf}_X(q) = ??, \quad \text{where } x = \mathrm{ibeta\_inv}(m/2, n/2, q).\]

Here \(\mathrm{ibeta\_inv}(\cdot)\) denotes the inverse of the real normalised incomplete beta function.

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

dist_beta_prime.isf(q)

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

\[\text{isf}_X(q) = ??, \quad \text{where } x = \mathrm{ibetac\_inv}(m/2, n/2, q).\]

Here \(\mathrm{ibetac\_inv}(\cdot)\) denotes the inverse of the real normalised complementary incomplete beta function.

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

dist_beta_prime.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a beta-prime distribution:

\[C_X(t) = ??\]

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

dist_beta_prime.m_x(t)

Returns NaN, since the moment generating function does not exist.

dist_beta_prime.k_x(t, k=0)

Returns NaN, since the cumulant generating function does not exist.

dist_beta_prime.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a beta-prime distribution. For \(-\alpha <k<\beta\) , the k-th moment is given by

\[\mu'_X(r) = \frac {B(\alpha +k,\beta -k)}{B(\alpha ,\beta )}.\]

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

dist_beta_prime.cumulants(k)

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

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