Boost: Arcsine Distribution

The following functions return the pdf, cdf, qtf or boost class of the arcsine distribution with endpoints \(a \in \mathbb{R}\) and \(b \in \mathbb{R}\), and the support interval \((a, b)\).

See also Wikipedia [1237], BoostMath [56].

Ctx.arcsine_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 arcsine distribution:

\[\text{pdf}(x) = \frac{1}{\pi \sqrt{(x-a)(b-x)}}.\]

The following example shows both forms of the syntax:

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

Ctx.arcsine_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 arcsine distribution:

\[\text{cdf}(x) = \frac{2}{\pi} \arcsin \left(\sqrt{\frac{x-a}{b-a}} \right)\]

The following example shows both forms of the syntax:

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

Ctx.arcsine_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 arcsine distribution:

\[\text{qtf}(q) = a + (b-a) \left( \sin \left( \tfrac{1}{2} \pi \cdot q\right) \right)^2\]

The following example shows both forms of the syntax:

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

class ctx.dist_arcsine(a=0, b=1)

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

The arcsine distribution is a continuous probability distribution with endpoints \(a \in \mathbb{R}\) and \(b \in \mathbb{R}\), \(a<b\), and the support interval \((a, b)\).

See also Wikipedia [1237], MathWorld [250], BoostMath [56], Witkovský [1613].

dist_arcsine.pdf(x)

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

\[\text{pdf}_X(x) = \frac{1}{\pi \sqrt{(x-a)(b-x)}}.\]

An example:

>>> from mpfunlab import mpm
>>> mpm.dps = 30; a = '0'; b = '1'; x = '0.25'
>>> print ("pdf: ", mpm.dist_arcsine(a, b).pdf(x))
6.3563523462564525615615615614561356E-20

dist_arcsine.cdf(x)

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

\[\text{cdf}_X(x) = \frac{2}{\pi} \arcsin \left(\sqrt{\frac{x-a}{b-a}} \right).\]

An example:

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

dist_arcsine.sf(x)

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

\[\text{sf}_X(x) = \frac{2}{\pi} \arccos \left(\sqrt{\frac{x-a}{b-a}} \right).\]

An example:

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

dist_arcsine.qtf(q)

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

\[\text{qtf}_X(q) = a + (b-a) \left( \sin \left( \tfrac{1}{2} \pi \cdot q\right) \right)^2\]

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

dist_arcsine.isf(q)

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

\[\text{isf}_X(q) = a + (b-a) \left( \cos \left( \tfrac{1}{2} \pi \cdot q\right) \right)^2\]

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

dist_arcsine.c_x(t)

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

\[C_X(t) = {}_1F_1\left( \frac{1}{2}; 1; it \right).\]

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

dist_arcsine.m_x(t)

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

\[M_X(t) = {}_1F_1\left( \frac{1}{2}; 1; t \right).\]

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

dist_arcsine.k_x(t, k=0)

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

\[K_X(t) = \log \left[ {}_1F_1\left( \frac{1}{2}; 1; t \right) \right].\]

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

dist_arcsine.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an arcsine distribution:

\[\mu'_{X}(r) = \prod_{j=0}^{n-1} \frac{2j+1}{2j+2}.\]

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

dist_arcsine.cumulants(k)

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

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