Trapezoidal Distribution
X ~ Trapezoidal(a, b, c, d)
Bounded / Risk-Modeling Family • Published July 22, 2026
Sits directly between the already-published Uniform and Triangular distributions and contains both as exact limiting cases (confirmed numerically to 2.8e-17 agreement at the Triangular limit); mean and second-moment closed forms derived via symbolic integration; maximum likelihood of the two interior breakpoints has no closed form, verified via 2D grid search showing convergence to the true values as the sample size grows.
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About This Reference Sheet
Sits directly between the already-published Uniform and Triangular distributions and contains both as exact limiting cases (confirmed numerically to 2.8e-17 agreement at the Triangular limit); mean and second-moment closed forms derived via symbolic integration; maximum likelihood of the two interior breakpoints has no closed form, verified via 2D grid search showing convergence to the true values as the sample size grows.
Support
x ∈ [a, d]
Parameters
a<b≤c<d (rise-start, flat-start, flat-end, fall-end)