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DISTRIBUTIONS / CONTINUOUS / PERT

Pert distribution

What it represents

PERT is a Beta distribution fitted to three estimates: minimum a, most likely b, and maximum c. It translates expert judgement or project scenarios into a bounded smooth distribution.

Historical clue

PERT, the Program Evaluation and Review Technique, popularized the distribution in U.S. Navy project planning during the 1950s. Beta-PERT was a practical approximation for uncertain activity times.

Relationships that clarify its use

It is a Beta parameterization, not a wholly separate family. Triangular also uses minimum, mode, and maximum but has linear shape; PERT smooths it with Beta exponents.

Data examples

  • project-activity durations and schedules
  • cost, effort, or time estimates with optimistic, likely, and pessimistic scenarios

Modelling warning

Do not treat b as the mean: it is the mode. The mean depends on the exponents and therefore on the chosen PERT convention.

Three estimates do not create evidence by themselves

PERT emerged from planning the Polaris project and turned optimistic, most likely, and pessimistic estimates into a Beta approximation for activity duration. In 1962 Clark explained the origin of formulas that had confused many analysts.

The usual version gives the mode four times the weight of either endpoint when forming the mean, while modern variants let that weight change. The three values are expert judgements, not an observed sample minimum, mode, and maximum. The distribution smooths an expert’s story but does not automatically include planning bias, dependence among tasks, or changes in the critical path. Across a network, adding PERT means does not reproduce the completion-time distribution.

Decision guide

A good candidate when: uncertainty is elicited through a minimum, most likely value, and maximum, and a smoother bounded curve than Triangular is desired.

Compare it with: Triangular and Beta. Perform sensitivity analysis on all three judgements and the mode weight; PERT summarizes opinions but does not turn weak estimates into data.

References

Pert Distribution: equations and calculator

Distribution defintion

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

Distribution domain

x∈[a,c]x\in\left[a,c\right]

Parameters domain and parameters constraints

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

Cumulative distribution function

FX(x)=I(z(x),α1,α2)F_{X}\left(x\right)=I(z(x),\alpha_{1},\alpha_{2})

Probability density function

fX(x)=(x−a)α1−1(c−x)α2−1Beta(α1,α2)(c−a)α1+α2−1f_{X}\left(x\right)=\frac{(x-a)^{\alpha_{1}-1}(c-x)^{\alpha_{2}-1}} {\text{Beta}(\alpha_{1},\alpha_{2})(c-a)^{\alpha_{1}+\alpha_{2}-1}}

Percent point function/Sample

FX−1(u)=a+(c−a)⋅I−1(u,α1,α2)F^{-1}_{X}\left(u\right)=a+(c-a)\cdot I^{-1}\left(u,\alpha_{1},\alpha_{2}\right)

Non-central parametric moments

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

Parametric mean

Mean(X)=μ1′=a+4b+c6\mathrm{Mean}(X)=\mu'_{1}=\frac{a+4b+c}{6}

Parametric variance

Variance(X)=μ2′−μ1′2=(Mean(X)−a)(c−Mean(X))7\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{(\mathrm{Mean}(X)-a)(c-\mathrm{Mean}(X))}{7}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2 (α2−α1)α1+α2+1(α1+α2+2)α1α2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{2\,(\alpha_{2}-\alpha_{1})\sqrt{\alpha_{1}+\alpha_{2}+1}}{(\alpha_{1}+\alpha_{2}+2)\sqrt{\alpha_{1}\alpha_{2}}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=6[(α1−α2)2(α1+α2+1)−α1α2(α1+α2+2)]α1α2(α1+α2+2)(α1+α2+3)+3\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}}=\frac{6[(\alpha_{1}-\alpha_{2})^2 (\alpha_{1} +\alpha_{2}+1)-\alpha_{1} \alpha_{2} (\alpha_{1}+\alpha_{2}+2)]}{\alpha_{1} \alpha_{2} (\alpha_{1}+\alpha_{2}+2) (\alpha_{1}+\alpha_{2}+3)}+3

Parametric median

Median(X)=a+(c−a)⋅I−1(12,α1,α2)\mathrm{Median}(X)=a+(c-a)\cdot I^{-1}\left(\frac{1}{2},\alpha_{1},\alpha_{2}\right)

Parametric mode

Mode(X)=b\mathrm{Mode}(X)=b

Additional information and definitions

z(x)=(x−a)/(c−a)z\left(x\right)=\left(x-a\right)/\left(c-a\right)
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
α1=4b+c−5ac−a,α2=5c−a−4bc−a\alpha_{1}=\frac{4b+c-5a} {c-a},\alpha_{2}=\frac{5c-a-4b} {c-a}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
I−1(x,a,b):Inverse of regularized incomplete beta functionI^{-1}\left(x,a,b\right):\text{Inverse of regularized incomplete beta function}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}