Boost: Weibull (Minimum-Type-III) distribution

The following functions return the pdf, cdf, qtf or boost class of the Weibull distribution with shape \(a > 0\), scale \(b > 0\), and the support interval \((0, +\infty)\).

See also Wikipedia [1286], MathWorld [912], BoostMath [96], Ehrhardt [309] (3.9.33).

Ctx.weibull_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 Weibull distribution:

\[\text{pdf}(x) = \frac{a}{x} \left(- \frac{x}{b}\right)^a \exp(-(x/b)^a).\]

The following example shows both forms of the syntax:

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

Ctx.weibull_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 Weibull distribution:

\[\text{cdf}(x) = 1 - \exp \left(- (x/b)^a\right) = -\text{expm1} \left(- (x/b)^a\right).\]

The following example shows both forms of the syntax:

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

Ctx.weibull_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 Weibull distribution:

\[\text{qtf}(q) = b \cdot \left(- \text{log1p}(-q)\right)^{1/a}.\]

The following example shows both forms of the syntax:

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

class ctx.dist_weibull(a, b)

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

The Weibull distribution is a continuous probability distribution with shape \(a > 0\), scale \(b > 0\), and the support interval \((0, +\infty)\). See also Wikipedia [1286], MathWorld [912], BoostMath [96], Witkovský [1632], R (Statistical System) [563].

dist_weibull.pdf(x)

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

\[\text{pdf}_X(x) = \frac{a}{x} \left(- \frac{x}{b}\right)^a \exp(-(x/b)^a).\]

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

dist_weibull.cdf(x)

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

\[\text{cdf}_X(x) = 1 - \exp \left(- (x/b)^a\right) = -\text{expm1} \left(- (x/b)^a\right).\]

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

dist_weibull.sf(x)

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

\[\text{sf}_X(x) = \exp \left(- (x/b)^a\right).\]

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

dist_weibull.qtf(q)

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

\[\text{qtf}_X(q) = b \cdot \left(- \text{log1p}(-q)\right)^{1/a}.\]

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

dist_weibull.isf(q)

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

\[\text{isf}_X(q) = b \cdot \left(- \log(q)\right)^{1/a}.\]

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

dist_weibull.c_x(t)

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

\[C_X(t) = \sum_{n=0}^{\infty} \frac{(it)^n \lambda^n}{n!} \Gamma(1+n/k).\]

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

dist_weibull.m_x(t)

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

\[M_X(t) = \sum_{n=0}^{\infty} \frac{t^n \lambda^n}{n!} \Gamma(1+n/k), k \ge 1.\]

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

dist_weibull.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a Weibull 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: ", weibull(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_weibull.moments(k)

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

\[\mu'_{X}(r) = \sum_{j=0}^{r} \binom{r}{j} \Gamma \left( \frac{r-j}{k} +1 \right) \lambda^{r-j}.\]

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

dist_weibull.cumulants(k)

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

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