Trapezoidal Distribution#
Two shape parameters giving the distances to the first and second modes 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 x\in\left[0,1\right].
\begin{eqnarray*}
u(c, d) & = & \frac{2}{d - c + 1} \\
f\left(x;c, d\right) & = & \left\{
\begin{array}{ccc}
\frac{ux}{c} & & x < c \\
u & & c\leq x \leq d \\
u\frac{1-x}{1-d} & & x > d \\
\end{array}
\right.\\
F\left(x;c, d\right) & = & \left\{
\begin{array}{ccc}
\frac{ux^{2}}{2c} & & x < c \\
\frac{uc}{2} + u(x-c) & & c\leq x \leq d \\
1 - \frac{u(1 - x)^2}{2(1 - d)} & & x > d \\
\end{array}
\right.\\
G\left(q;c, d\right) & = & \left\{
\begin{array}{ccc}
\sqrt{qc(d-c+1)} & & q < c \\
\frac{q}{u}+ \frac{c}{2} & & q \leq d \\
1 - \sqrt{\frac{2(1 - q) (1 - d)}{u}} & & q > d \\
\end{array}
\right.
\end{eqnarray*}
Implementation: scipy.stats.trapezoid