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DISTRIBUTIONS / CONTINUOUS / TRIANGULAR

Triangular distribution

The idea in one sentence

The Triangular distribution represents uncertainty with minimum a, most-likely value b, and maximum c. Its two straight-line pieces are easy to explain when few parameters are available.

Support and interpretation

Support is [a,c]; b is the mode. With b at the midpoint it is symmetric; moving it creates skewness toward one boundary.

Its place in the family

PERT replaces straight lines with a smooth Beta shape. Triangular is a limiting trapezoidal form and relates to Beta when exponents produce linear slopes.

An applied reading

It is a useful first approximation when bounds and a mode are known but data are insufficient for parametric estimation. Sums of many triangular activities can become nearly normal.

Bounds are often subjective and three points do not encode dependence between activities; document both assumptions.

The distribution that draws exactly what was stated

A minimum, mode, and maximum determine two straight lines. There is no hidden tail beyond the endpoints and no further curvature to estimate. That transparency explains its popularity in simulation when only expert judgement is available.

The symmetric case has an interesting construction: averaging two independent Uniform variables gives a Bates law with n=2, which is Triangular. The difference of two equal Uniform variables also creates a centred triangular shape. These stories do not automatically justify a subjective mode, but they show where the law appears mechanically. Compared with PERT, Triangular places relatively more mass near the bounds. That choice can change aggregate risk even with identical three-point estimates.

Decision guide

A good candidate when: information is limited but a finite minimum, mode, and maximum can be defended.

Compare it with: PERT for a smoother shape and Beta when data are available. Include sensitivity analysis: the three points are usually judgements, not perfectly known limits.

References

Triangular Distribution: equations and calculator

Distribution defintion

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

Distribution domain

x∈[a,b]x\in\left[a,b\right]

Parameters domain and parameters constraints

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

Cumulative distribution function

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

Probability density function

fX(x)={2(x−a)(b−a)(c−a)if  a≤x<c,2(b−x)(b−a)(b−c)if  c≤x≤b,f_{X}\left(x\right)=\left\{\begin{array}{cl}\frac{2(x-a)}{(b-a)(c-a)} & \text{if } \ a \leq x < c,\\ \frac{2(b-x)}{(b-a)(b-c)} & \text{if } \ c \leq x \leq b,\\ \end{array} \right.

Percent point function/Sample

FX−1(u)={a+U(b−a)(c−a)if  0<U<c−ab−ab−(1−U)(b−a)(b−c)if  c−ab−a≤U<1F^{-1}_{X}\left(u\right)=\left\{\begin{array}{cl} a+\sqrt{U(b-a)(c-a)} & \text{if } \ 0 < U < \frac{c-a}{b-a} \\ b-\sqrt{(1-U)(b-a)(b-c)} & \text{if } \ \frac{c-a}{b-a} \leq U < 1 \end{array} \right.

Non-central parametric moments

μk′=E[Xk]=∫abxkfX(x)dx\mu'_{k}=E[X^k]=\int_{a}^{b}x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=a+b+c3\mathrm{Mean}(X)=\mu'_{1}=\frac{a+b+c}{3}

Parametric variance

Variance(X)=μ2′−μ1′2=a2+b2+c2−ab−ac−bc18\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{a^2+b^2+c^2-ab-ac-bc}{18}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2(a ⁣+ ⁣b ⁣− ⁣2c)(2a ⁣− ⁣b ⁣− ⁣c)(a ⁣− ⁣2b ⁣+ ⁣c)5(a2 ⁣+ ⁣b2 ⁣+ ⁣c2 ⁣− ⁣ab ⁣− ⁣ac ⁣− ⁣bc)32\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{\sqrt 2 (a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^2\!+\!b^2\!+\!c^2\!-\!ab\!-\!ac\!-\!bc)^\frac{3}{2}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3−35\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}}=3-\frac{3}{5}

Parametric median

Median(X)={a+(b−a)(c−a)2if c≥a+b2b−(b−a)(b−c)2if c≤a+b2\mathrm{Median}(X)=\left\{\begin{array}{cl} a+\sqrt{\frac{(b-a)(c-a)}{2}} & \text{if } c \ge\frac{a+b}{2} \\ b-\sqrt{\frac{(b-a)(b-c)}{2}} & \text{if } c \le\frac{a+b}{2} \end{array} \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}