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Trapezoidal distribution

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The Trapezoidal distribution is bounded with density that rises linearly, may stay flat, and then falls linearly. It represents interval uncertainty with a plausible central range rather than one most-likely point.

Feature Reading
Support Four points a ≤ b ≤ c ≤ d define the support and plateau. Sloped sections soften the jumps of uniform or triangular shapes.
Shape At b=c it becomes triangular; with a=b or c=d one side is triangular. Uniform is the limiting case where the density remains flat throughout.

Origin and terminology

It is used in risk analysis, simulation, and possibility theory as a simple representation of imprecise knowledge. It has no single historical origin.

In real models

  • project estimates and interval-plus-plateau risk analysis
  • Monte Carlo simulation when experts specify minimum, likely range, and maximum

Comparisons

Do not confuse a trapezoidal density with a fuzzy membership function: one integrates to 1 and the other need not.

When an expert prefers a plateau to one mode

Triangular forces a single most likely value. Trapezoidal allows an entire central interval to be equally plausible: density rises from the minimum, stays flat between two points, and falls toward the maximum. Uniform appears when the plateau fills the support, while Triangular appears when it collapses to one point.

The geometry helps in risk workshops because every breakpoint is visible. Four subjective points are not four precise estimates, however. A broad plateau can express weak information, but its linear edges remain concrete claims. In project simulation, vary the points and measure how output quantiles respond, especially when several inputs share the same expert judgement.

Decision guide

A good candidate when: only bounds and a central interval of equally plausible values are known, with plausibility changing linearly on the flanks.

Compare it with: Triangular for a single most-likely value and Uniform for no central preference. Corners encode elicited judgement and do not automatically describe a smooth observed process.

References

Trapezoidal Distribution: equations and calculator

Distribution defintion

X∼Trapezoidal(a,b,c,d)X\sim\mathrm{Trapezoidal}\left(a,b,c,d\right)

Distribution domain

x∈[a,d]x\in\left[a,d\right]

Parameters domain and parameters constraints

a∈R,b∈R,c∈R,d∈R,a<b<c,b<c<da\in\mathbb{R},b\in\mathbb{R},c\in\mathbb{R},d\in\mathbb{R},a < b < c,b < c < d

Cumulative distribution function

FX(x)={1d+c−a−b1b−a(x−a)2if  a≤x<b1d+c−a−b(2x−a−b)if  b≤x<c1−1d+c−a−b1d−c(d−x)2if  c≤x≤dF_{X}\left(x\right)=\left\{\begin{array}{cl}\frac{1}{d+c-a-b}\frac{1}{b-a}(x-a)^2 & \text{if } \ a\leq x < b \\ \frac{1}{d+c-a-b}(2x-a-b) & \text{if } \ b\leq x < c \\ 1-\frac{1}{d+c-a-b}\frac{1}{d-c}(d-x)^2 & \text{if } \ c\leq x \le d \end{array} \right.

Probability density function

fX(x)={2d+c−a−bx−ab−aif  a≤x<b2d+c−a−bif  b≤x<c2d+c−a−bd−xd−cif  c≤x≤df_{X}\left(x\right)=\left\{\begin{array}{cl}\frac{2}{d+c-a-b}\frac{x-a}{b-a} & \text{if } \ a\leq x < b \\ \frac{2}{d+c-a-b} & \text{if } \ b\leq x < c \\ \frac{2}{d+c-a-b}\frac{d-x}{d-c} & \text{if } \ c\leq x \leq d \end{array} \right.

Percent point function/Sample

FX−1(u)={a+u×(d+c−a−b)×(b−a)if u≤A1(a+b+u×(d+c−a−b))/2if A1≤u≤A1+A2d−(1−u)×(d+c−a−b)×(d−c)if A1+A2≤u≤A1+A2+A3F^{-1}_{X}\left(u\right)=\left\{\begin{array}{cl} a+\sqrt{u\times (d+c-a-b)\times (b-a)} & \text{if } u \leq A_{1} \\ (a+b+u\times (d+c-a-b))/2 & \text{if } A_{1} \leq u \leq A_{1}+A_{2} \\ d-\sqrt{(1-u)\times (d+c-a-b)\times (d-c)} & \text{if } A_{1}+A_{2} \leq u \leq A_{1}+A_{2}+A_{3} \end{array} \right.

Non-central parametric moments

μk′=E[Xk]=∫abxkfX(x)dx=2d+c−b−a1(k+1)(k+2)(dk+2−ck+2d−c−bk+2−ak+2b−a)\mu'_{k}=E[X^k]=\int_{a}^{b}x^{k}f_{X}\left(x\right)dx=\frac{2}{d+c-b-a}\frac{1}{(k+1)(k+2)}\left(\frac{d^{k+2}-c^{k+2}}{d-c}-\frac{b^{k+2}-a^{k+2}}{b-a}\right)

Parametric mean

Mean(X)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Parametric variance

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Parametric median

Median(X)=FX−1(1/2)\mathrm{Median}(X)=F^{-1}_{X}\left(1/2\right)

Parametric mode

Mode(X)∈[b,c]\mathrm{Mode}(X)\in [b,c]

Additional information and definitions

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
A1=(b−a)/(d+c−a−b)A_{1}=(b-a)/(d+c-a-b)
A2=2(c−b)/(d+c−a−b)A_{2}=2(c-b)/(d+c-a-b)
A3=(d−c)/(d+c−a−b)A_{3}=(d-c)/(d+c-a-b)