.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| .. _rst_dist_triangular: Boost: Triangular Distribution ------------------------------------------------------------------------------- The following functions return the pdf, cdf, qtf or boost class of the triangular distribution, with finite `a`) of the triangular distribution: .. math:: \text{pdf}(x) = \begin{cases} 0 & xb \end{cases} The following example shows both forms of the syntax: .. code-block:: pycon >>> 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 |cr| .. _Ctx_TriangularCdf: .. method:: Ctx.triangular_cdf(x, lower, mode, upper) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{cdf}(x)`, the value of the cumulative distribution function (:ref:`Cdf `) of the triangular distribution: .. math:: \text{cdf}(x) = \begin{cases} 0 & xb \end{cases} The following example shows both forms of the syntax: .. code-block:: pycon >>> 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 |cr| .. _Ctx_TriangularQtf: .. method:: Ctx.triangular_qtf(q, lower, mode, upper) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{qtf}(q)`, the value of the quantile function (:ref:`Qtf `) of the triangular distribution: .. math:: \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}, The following example shows both forms of the syntax: .. code-block:: pycon >>> 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 |cr| .. py: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 `ab \end{cases} .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("pdf: ", triangular(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_triangular.cdf(x) Returns `\text{cdf}_X(x)`, the cumulative distribution function (cdf) of a random variable `X`, following a triangular distribution: .. math:: \text{cdf}_X(x) = \begin{cases} 0 & xb \end{cases} .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("cdf: ", triangular(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_triangular.sf(x) Returns `\text{sf}_X(x)`, the survival function (sf) of a random variable `X`, following a triangular distribution: .. math:: \text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print (" sf: ", triangular(mu, sigma).pdf(x)) sf: 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_triangular.qtf(q) Returns `\text{qtf}_X(x)`, the quantile function (qtf) of a random variable `X`, following a triangular distribution: .. math:: \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}, .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("qtf: ", triangular(mu, sigma).qtf(q)) qtf: 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_triangular.isf(q) Returns `\text{isf}_X(q)`, the inverse survival function (isf) of a random variable `X`, following a triangular distribution: .. math:: \text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("isf: ", triangular(mu, sigma).isf(q)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_triangular.c_x(t) Returns `C_X(t)`, the characteristic function of a random variable `X`, following a triangular distribution: .. math:: 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}. .. code-block:: python >>> 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 |cr| .. method:: dist_triangular.m_x(t) Returns `M_X(t)`, the moment generating function of a random variable `X`, following a triangular distribution: .. math:: 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} .. code-block:: python >>> 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 |cr| .. method:: dist_triangular.k_x(t, k = 0) Returns `K_X(t)`, the cumulant generating functionof a random variable `X`, following a triangular distribution: .. math:: 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]. .. code-block:: python >>> 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 |cr| .. method:: 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: .. math:: \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)} .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; k = 6; >>> print ("saddlepoint: ", triangular(mu, sigma).moments(k)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: 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. .. code-block:: python >>> 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.)