Boost: Uniform distribution

Returns the pdf, cdf, qtf or boost class of a random variable \(X\), following a uniform distribution on the support interval \([a, b]\) with finite \(a < b\).

See also Wikipedia [1285], MathWorld [911], BoostMath [95], Team [562], Ehrhardt [309] (3.9.31).

Ctx.uniform_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 uniform distribution:

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

The following example shows both forms of the syntax:

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

Ctx.uniform_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 uniform distribution:

\[\text{cdf}(x) = \frac{x-a}{b-a}.\]

The following example shows both forms of the syntax:

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

Ctx.uniform_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 uniform distribution:

\[\text{qtf}(q) = a+q(b-a).\]

The following example shows both forms of the syntax:

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

class ctx.dist_uniform(lower, upper)

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

The uniform distribution is a continuous probability distribution on the support interval \([a, b]\) with finite \(a < b\). See also Wikipedia [1285], MathWorld [911], BoostMath [95], Witkovský [1629], R (Statistical System) [562].

dist_uniform.pdf(x)

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

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

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

dist_uniform.cdf(x)

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

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

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

dist_uniform.sf(x)

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

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t.\]

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

dist_uniform.qtf(q)

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

\[\text{qtf}_X(q) = \frac{x-a}{b-a}.\]

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

dist_uniform.isf(q)

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

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

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

dist_uniform.c_x(t)

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

\[C_X(t) = \frac{e^{itb} - e^{ita}}{it(b-a)}, \quad \text{for } t\ne 0, 0 \text{ otherwise}.\]

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

dist_uniform.m_x(t)

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

\[M_X(t) = \frac{e^{tb} - e^{ta}}{t(b-a)}, \quad \text{for } t\ne 0, 1 \text{ otherwise}.\]

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

dist_uniform.k_x(t, k=0)

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

\[K_X(t) = K_X(t) = \log(M_X(t)).\]

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

dist_uniform.moments(k)

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

\[\mu'_{X}(n) = \frac{1}{n+1} \sum_{k=0}^{n} a^k b^{n-k}.\]

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

dist_uniform.cumulants(k)

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

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