!!!Boost: Kumaraswamy distribution#

The following functions return pdf, cdf, qtf or boost class of the Kumaraswamy distribution with shape parameters \(a > 0\), \(b > 0\), and the support interval \((0, 1)\).

See also Wikipedia [1275], MathWorld [259], Ehrhardt [309] (3.9.13).

Ctx.kumaraswamy_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 Kumaraswamy distribution:

\[\text{pdf}(x) = a b x^{a-1}{(1-x^{a})}^{b-1} = a b x^{a-1} \cdot (-\text{powm1}(x,a))^{b-1}.\]

The following example shows both forms of the syntax:

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

Ctx.kumaraswamy_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 Kumaraswamy distribution:

\[\text{cdf}(x) = 1 - (1-x^{a})^{b} = -\text{powm1}(-\text{powm1}(x,a), b).\]

The following example shows both forms of the syntax:

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

Ctx.kumaraswamy_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 Kumaraswamy distribution:

\[\text{qtf}(q) = (1-(1-q)^{\frac {1}{b}})^{\frac {1}{a}} = \text{pow}(-\text{pow1pm1}(-q, 1/b), 1/a).\]

The following example shows both forms of the syntax:

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

class ctx.dist_kumaraswamy(a, b)#

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

The Kumaraswamy distribution is a continuous probability distribution with shape parameters \(a > 0\), \(b > 0\), and the support interval \((0, 1)\). See also Wikipedia [1275], MathWorld [259].

dist_kumaraswamy.pdf(x)#

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

\[\text{pdf}_X(x) = a b x^{a-1}{(1-x^{a})}^{b-1} = a b x^{a-1} \cdot (-\text{powm1}(x,a))^{b-1}.\]

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

dist_kumaraswamy.cdf(x)#

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

\[\text{cdf}_X(x) = 1 - (1-x^{a})^{b} = -\text{powm1}(-\text{powm1}(x,a), b).\]

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

dist_kumaraswamy.sf(x)#

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

\[\text{sf}_X(x) = (1-x^{a})^{b} = \text{pow}(-\text{powm1}(x,a), b).\]

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

dist_kumaraswamy.qtf(q)#

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

\[\text{qtf}_X(q) = (1-(1-q)^{\frac {1}{b}})^{\frac {1}{a}} = \text{pow}(-\text{pow1pm1}(-q, 1/b), 1/a).\]

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

dist_kumaraswamy.isf(q)#

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

\[\text{isf}_X(q) = (1-q^{\frac {1}{b}})^{\frac {1}{a}} = \text{pow}(-\text{powm1}(q, 1/b), 1/a).\]

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

dist_kumaraswamy.c_x(t)#

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

\[C_X(t) = tbd\]

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

dist_kumaraswamy.m_x(t)#

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

\[M_X(t) = tbd\]

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

dist_kumaraswamy.k_x(t, k=0)#

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

\[K_X(t) = tbd\]

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

dist_kumaraswamy.moments(k)#

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

\[\mu_{X}(r) = {\frac {b\Gamma (1+n/a)\Gamma (b)}{\Gamma (1+b+n/a)}}=bB(1+n/a,b).\]

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

dist_kumaraswamy.cumulants(k)#

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

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

!!!Boost: Kumaraswamy distribution

Contents

!!!Boost: Kumaraswamy distribution#