Boost: Triangular Distribution

The following functions return the pdf, cdf, qtf or boost class of the triangular distribution, with finite \(a<b\), mode \(c, a \le c \le b\), and the support interval \([a, b]\).

See also Wikipedia [1284], MathWorld [910], BoostMath [94], Ehrhardt [309] (3.9.30).

Ctx.triangular_pdf(x, lower, mode, upper)

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the triangular distribution:

\[\begin{split}\text{pdf}(x) = \begin{cases} 0 & x<a\\ \frac{2(x-a)}{(b-a)(c-a)} & a \leq x < c\\ \frac{2}{b-a} & x = c\\ \frac{2(b-x)}{(b-a)(b-c)} & c < x \leq b\\ 0 & x>b \end{cases}\end{split}\]

The following example shows both forms of the syntax:

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

Ctx.triangular_cdf(x, lower, mode, upper)

where Ctx is Math53 or CtxBoost.

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

\[\begin{split}\text{cdf}(x) = \begin{cases} 0 & x<a\\ \frac{(x-a)^2}{(b-a)(c-a)} & a \leq x < c\\ \frac{c-a}{b-a} & x = c\\ 1-\frac{(b-x)^2}{(b-a)(b-c)} & c < x \leq b\\ 1 & x>b \end{cases}\end{split}\]

The following example shows both forms of the syntax:

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

Ctx.triangular_qtf(q, lower, mode, upper)

where Ctx is Math53 or CtxBoost.

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

\[\begin{split}\text{qtf}(q) = \begin{cases} a+\sqrt{(b-a)(c-a)y} & y<(c-a)/(b-a)\\ c & y=(c-a)/(b-a) \\ b-\sqrt{(b-a)(b-c)(1-y)} & y>(c-a)/(b-a) \end{cases},\end{split}\]

The following example shows both forms of the syntax:

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

class ctx.dist_triangular(lower, mode, upper)

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

The triangular distribution is a continuous probability distribution with finite \(a<b\), mode \(c, a \le c \le b\), and the support interval \([a, b]\). See also Wikipedia [1284], MathWorld [910], BoostMath [94], Witkovský [1631].

dist_triangular.pdf(x)

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

\[\begin{split}\text{pdf}_X(x) = \begin{cases} 0 & x<a\\ \frac{2(x-a)}{(b-a)(c-a)} & a \leq x < c\\ \frac{2}{b-a} & x = c\\ \frac{2(b-x)}{(b-a)(b-c)} & c < x \leq b\\ 0 & x>b \end{cases}\end{split}\]

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

dist_triangular.cdf(x)

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

\[\begin{split}\text{cdf}_X(x) = \begin{cases} 0 & x<a\\ \frac{(x-a)^2}{(b-a)(c-a)} & a \leq x < c\\ \frac{c-a}{b-a} & x = c\\ 1-\frac{(b-x)^2}{(b-a)(b-c)} & c < x \leq b\\ 1 & x>b \end{cases}\end{split}\]

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

dist_triangular.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a triangular 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: ", triangular(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_triangular.qtf(q)

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

\[\begin{split}\text{qtf}_X(q) = \begin{cases} a+\sqrt{(b-a)(c-a)y} & y<(c-a)/(b-a)\\ c & y=(c-a)/(b-a) \\ b-\sqrt{(b-a)(b-c)(1-y)} & y>(c-a)/(b-a) \end{cases},\end{split}\]

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

dist_triangular.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a triangular 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: ", triangular(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_triangular.c_x(t)

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

\[C_X(t) = 2 \frac{(b-c) e^{iat} - (b-a)e^{ict} + (c-a)e^{ibt}}{(b-a)(c-a)(b-c)t^2}.\]

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

dist_triangular.m_x(t)

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

\[M_X(t) = 2 \frac{(b-c) e^{at} - (b-a)e^{ct} + (c-a)e^{bt}}{(b-a)(c-a)(b-c)t^2}\]

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

dist_triangular.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating functionof a random variable \(X\), following a triangular distribution:

\[K_X(t) = \log \left[ 2 \frac{(b-c) e^{at} - (b-a)e^{ct} + (c-a)e^{bt}}{(b-a)(c-a)(b-c)t^2} \right].\]

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

dist_triangular.moments(k)

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

\[\mu'_{n+1} = \sum_{i=0}^{k} \binom{k}{i}(b-a)^i a^{k-1} \frac{2(1-\theta^{i+1})}{(i+1)(i+2)(1-\theta)}\]

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

dist_triangular.cumulants(k)

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

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

References:

Kotz Triangular (moments.)