Trapezoidal Distribution

Two shape parameters \(c\in[0,1], d\in[0, 1]\) 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