Boost: Binomial distribution

Boost: Binomial distribution#

The following functions return the pmf, cdf, qtf or boost class of the binomial distribution with number of trials \(n \ge 0\) and success probability \(0 \le p \le 1\), and \(0 \le q \le 1\).

See also Wikipedia [1265], MathWorld [889], BoostMath [77], Ehrhardt [309] (3.9.3).

Ctx.binomial_pmf(k, n, p)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{pmf}(x)\), the value of the probability density function (Pmf) of the binomial distribution:

\[\text{pmf}(x) = \binom{n}{k} p^k (1-p)^{n-k} = f_{\text{Beta}}(k+1,n-k+1,p)/(n+1).\]

Here \(f_{\text{Beta}}(\cdot)\) denotes the PDF of the central beta distribution.

The following example shows both forms of the syntax:

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

Ctx.binomial_cdf(k, n, p)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the binomial distribution:

\[\text{cdf}(x) = \sum_{j=0}^{k} \text{pmf}_X(j) = \text{ibetac}(k+1, n-k, p).\]

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

The following example shows both forms of the syntax:

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

Ctx.binomial_qtf(q, n, p)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the binomial distribution:

There is no known closed form for \(\text{qtf}(q)\): it is computed with Newton iterations where the starting values are from the corresponding Boost functions.

The following example shows both forms of the syntax:

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

class ctx.dist_binomial(n, p)#

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

The binomial distribution is a discrete (lattice) probability distribution with number of trials \(n \ge 0\) and success probability \(0 \le p \le 1\). See also Wikipedia [1265], MathWorld [889], BoostMath [77], Witkovský [1633], R (Statistical System) [556].

A special case is the Bernoulli distribution, see See also Wikipedia [1264], MathWorld [888], and BoostMath [76].

dist_binomial.pmf(x)#

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following an binomial distribution:

\[\text{pmf}_X(x) = \binom{n}{k} p^k (1-p)^{n-k} = f_{\text{Beta}}(k+1,n-k+1,p)/(n+1).\]

and \(f_{\text{Beta}}(\cdot)\) and \(F_{\text{Beta}}(\cdot)\) denote the PDF and CDF, respectively, of the central beta distribution.

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

dist_binomial.cdf(x)#

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

\[\text{cdf}_X(x) = \sum_{j=0}^{k} \text{pmf}_X(j) = F_{\text{Beta}}(1-p; n-k, k+1).\]

and \(f_{\text{Beta}}(\cdot)\) and \(F_{\text{Beta}}(\cdot)\) denote the PDF and CDF, respectively, of the central beta distribution.

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

dist_binomial.sf(x)#

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

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x).\]

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

dist_binomial.qtf(q)#

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following an binomial distribution. There is no closed form for the qtf: It is computed with Newton iterations where the starting values are from Boost.

\[\text{qtf}_X(q) = tbd.\]

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

dist_binomial.isf(q)#

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

\[\text{isf}_X(q) = \text{qtf}_X(1-q).\]

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

dist_binomial.g_x(t)#

Returns \(G_X(t)\), the probability generating function of a random variable \(X\), following an binomial distribution:

\[G_X(t) = \left(P t + Q\right)^n.\]

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

dist_binomial.c_x(t)#

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

\[C_X(t) = \left(P e^{it} + Q\right)^n.\]

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

dist_binomial.m_x(t)#

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

\[M_X(t) = \left(P e^{t} + Q\right)^n.\]

\[L_X(t) = \left(P e^{-t} + Q\right)^n.\]

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

dist_binomial.k_x(t, k=0)#

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

\[K_X(t) = K_X(t) = n \log \left(P e^t + Q\right).\]

\(K_X(t)\), 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 binomial distribution, are defined as

\[K_X(t) = n \log \left(p \cdot e^t + 1-p\right),\]

\[K_X^{(1)}(t) = \frac{n \cdot p \cdot e^x}{p(e^x-1)+1},\]

\[K_X^{(2)}(t) = -\frac{n (p-1) p \cdot e^x}{(p(e^x-1)+1)^2},\]

\[K_X^{(3)}(t) = \frac{n (p-1) p \cdot e^x ( p \cdot e^x +p-1)}{(p(e^x-1)+1)^3},\]

and for \(j \ge 4\) the derivatives are calculated by numerically differentiating \(K_X^{(3)}(t)\).

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

dist_binomial.moments(k)#

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

\[\mu^{}_X(r) = n P Q \sum_{i=0}^{r-2} \binom{r-1}{i} \mu_i - P \sum_{i=0}^{r-2} \binom{r-1}{i} \mu_{i+1}, \quad r>2; \quad \mu_1 = nP, \quad \mu_2 = nPQ.\]

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

dist_binomial.cumulants(k)#

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

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

Approximations

ctx.hypergeo_ft(x, n, results='cdf')#

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

Calculates the pdf, cdf and sf from the characteristic function (see pmf_from_cf_lattice() and cdf_from_cf_lattice()).

ctx.binomial_ft(x, n, results='cdf')#

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

Calculates the pdf, cdf and sf from the characteristic function (see pmf_from_cf_lattice() and cdf_from_cf_lattice()).

ctx.binomial_ecf(k, n, p, results='cdf')#

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

Calculates the Edgeworth approximation to the pdf, cdf and sf, support \(k \in \{0, \cdots n\}\).

ctx.binomial_ecf_inv(q, n, p, results='qtf')#

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

Calculates the Cornish-Fisher approximation to the qtf and isf.

ctx.binomial_spa(k, n, p, results='cdf')#

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

Calculates the Luggannini-Rice saddlepoint approximation of the pdf, cdf and sf.

The saddlepoint is given by

\[\hat{s}(x)= \log \left( \frac{k Q}{(n-k) P} \right).\]

ctx.binomial_spa_inv(q, n, p, results='qtf')#

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

Calculates the inverse Jensen saddlepoint approximation of the qtf and isf.

Boost: Binomial distribution

Contents

Boost: Binomial distribution#