Boost: Beta (Pearson Type I and II) distribution#

Returns the pdf, cdf, qtf or boost class of a random variable \(X\), following a beta distribution with parameters \(a > 0\), \(b > 0\), and the support interval \((0, 1)\).

See also Wikipedia [1238], MathWorld [868], BoostMath [57], Ehrhardt [309] (3.9.2).

Ctx.beta_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 beta distribution:

\[\text{pdf}(x) = f_{\text{Beta}}(a,b,x) = \frac{1}{B(a,b)} x^{a-1}(1-x)^{b-1}.\]

The following example shows both forms of the syntax:

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

Ctx.beta_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 beta distribution:

\[\text{cdf}(x) = I_x(a, b) = \text{ibeta}(a, b, x).\]

Here \(\text{ibeta}(\cdot)\) denotes the real normalised incomplete beta function (RealIBeta).

The following example shows both forms of the syntax:

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

Ctx.beta_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 beta distribution:

\[\text{qtf}(q) = \mathrm{ibeta\_inv}(a, b, q).\]

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

The following example shows both forms of the syntax:

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

class ctx.dist_beta(a, b)#

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

The beta distribution is a continuous probability distribution with parameters \(a > 0\), \(b > 0\), and the support interval \((0, 1)\). See also Wikipedia [1238], MathWorld [868], BoostMath [57], Witkovský [1614], R (Statistical System) [545].

dist_beta.pdf(x)#

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

\[\text{pdf}_X(x) = f_{\text{Beta}}(a,b,x) = \frac{1}{B(a,b)} x^{a-1}(1-x)^{b-1}.\]

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

dist_beta.cdf(x)#

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

\[\text{cdf}_X(x) = I_x(a, b) = \text{ibeta}(a, b, x).\]

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

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

dist_beta.sf(x)#

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

\[\text{sf}_X(x) = 1 - I_x(a, b) = \text{ibetac}(a, b, x).\]

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

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

dist_beta.qtf(q)#

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

\[\text{qtf}_X(q) = \mathrm{ibeta\_inv}(a, b, q).\]

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: ", beta(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_beta.isf(q)#

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

\[\text{isf}_X(q) = \mathrm{ibetac\_inv}(a, b, q).\]

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: ", beta(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_beta.c_x(t)#

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

\[C_X(t) = {}_1F_1 (a, a+b; it).\]

where \({}_1F_1()\) is Kummer’s confluent hypergeometric function (of the first kind).

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

dist_beta.m_x(t)#

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

\[M_X(t) = {}_1F_1 (a, a+b; t).\]

where \({}_1F_1()\) is Kummer’s confluent hypergeometric function (of the first kind).

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

dist_beta.k_x(t, k=0)#

Returns \(K_X(t)\), the cumulant generating function, and its \(j^{\text{th}}\) derivatives, \(K_X^{(j)}(t), j = 1 \ldots k\), of a random variable \(X\), following a beta distribution:

\[K_X(t) = \log ( {}_1F_1 (a, a+b; t) ),\]

\[K_X^{(j)}(t) = tbd .\]

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

dist_beta.moments(k)#

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a beta distribution: the moments are calculated from the cumulants.

\[\mu'_{h} = \frac{\Gamma(a+h)\Gamma(a+b)}{\Gamma(a)\Gamma(a+b+h)}\]

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

dist_beta.cumulants(k)#

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

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

Recurrences: Central Beta

ctx.beta_recurrence(x, n, lambda)#

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

The following recurrence relations hold for the pdf and CDF:

\begin{eqnarray} I(a,b;x) & = & 1-I(b,a;1-x) \\ I(a,b;x) & = & \binom{n}{a} x^a (1-x)^{b-1} + I(a+1,b-1; x) \\ I(a,b;x) & = & \binom{n}{a} x^a (1-x)^{b} + I(a+1,b; x) \\ I(a,b+1;x) & = & \binom{n}{a} x^a (1-x)^{b} + I(a,b; x) \\ I(a,b;x) & = & \binom{n}{a+b} x^a (1-x)^{b} \frac{a}{a+b-x} + I(a+1,b+1; x) \\ I(a,b;x) & = & F\left(2a,2b, \frac{nx}{m-mx}\right) \end{eqnarray}

Boost: Beta (Pearson Type I and II) distribution

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Boost: Beta (Pearson Type I and II) distribution#