1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
|
.. _continuous-triang:
Triangular Distribution
=======================
One shape parameter :math:`c\in[0,1]` giving the distance to the peak as a percentage of the total extent of
the non-zero portion. The location parameter is the start of the non-
zero portion, and the scale-parameter is the width of the non-zero
portion. In standard form we have :math:`x\in\left[0,1\right].`
.. math::
:nowrap:
\begin{eqnarray*}
f\left(x;c\right) & = & \left\{
\begin{array}{ccc}
2\frac{x}{c} & & x < c \\
2\frac{1-x}{1-c} & & x \geq c
\end{array}
\right.\\
F\left(x;c\right) & = & \left\{
\begin{array}{ccc}
\frac{x^{2}}{c} & & x < c \\
\frac{x^{2}-2x+c}{c-1} & & x \geq c
\end{array}
\right.\\
G\left(q;c\right) & = & \left\{
\begin{array}{ccc}
\sqrt{cq} & & q < c \\
1-\sqrt{\left(1-c\right)\left(1-q\right)} & & q \geq c
\end{array}
\right.
\end{eqnarray*}
.. math::
:nowrap:
\begin{eqnarray*} \mu & = & \frac{c}{3}+\frac{1}{3}\\ \mu_{2} & = & \frac{1-c+c^{2}}{18}\\ \gamma_{1} & = & \frac{\sqrt{2}\left(2c-1\right)\left(c+1\right)\left(c-2\right)}{5\left(1-c+c^{2}\right)^{3/2}}\\ \gamma_{2} & = & -\frac{3}{5}\end{eqnarray*}
.. math::
:nowrap:
\begin{eqnarray*} h\left(X\right) & = & \log\left(\frac{1}{2}\sqrt{e}\right)\\ & \approx & -0.19314718055994530942.\end{eqnarray*}
Implementation: `scipy.stats.triang`
|
