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{array}{rcl} u (c, d) & = & \frac{2}{d - c + 1} \\ f (x; c, d) & = & {\begin{array}{ccc} \frac{u x}{c} & x < c \\ u & c \leq x \leq d \\ u \frac{1 - x}{1 - d} & x > d \end{array} \\ F (x; c, d) & = & {\begin{array}{ccc} \frac{u x^{2}}{2 c} & x < c \\ \frac{u c}{2} + u (x - c) & c \leq x \leq d \\ 1 - \frac{u (1 - x)^{2}}{2 (1 - d)} & x > d \end{array} \\ G (q; c, d) & = & {\begin{array}{ccc} \sqrt{q c (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} \end{array}

$\begin{eqnarray*} u(c, d) & = & \frac{2}{d - c + 1} \\ f\left(x;c, d\right) & = & \left\{ \begin{array}{ccc} \frac{ux}{c} & & x < c \\ u & & c\leq x \leq d \\ u\frac{1-x}{1-d} & & x > d \\ \end{array} \right.\\ F\left(x;c, d\right) & = & \left\{ \begin{array}{ccc} \frac{ux^{2}}{2c} & & x < c \\ \frac{uc}{2} + u(x-c) & & c\leq x \leq d \\ 1 - \frac{u(1 - x)^2}{2(1 - d)} & & x > d \\ \end{array} \right.\\ G\left(q;c, d\right) & = & \left\{ \begin{array}{ccc} \sqrt{qc(d-c+1)} & & q < c \\ \frac{q}{u}+ \frac{c}{2} & & q \leq d \\ 1 - \sqrt{\frac{2(1 - q) (1 - d)}{u}} & & q > d \\ \end{array} \right. \end{eqnarray*}$

Implementation: scipy.stats.trapezoid