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PERT distribution
PERT distribution
PERT distribution
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PERT
Probability density function
Example density curves for the PERT probability distribution
Cumulative distribution function
Example cumulative distribution curves for the PERT probability distribution
Parameters (real)
(real)
Support
PDF

where

CDF

(the regularized incomplete beta function) with
Mean
Median

Mode
Variance
Skewness
Excess kurtosis

In probability and statistics, the PERT distributions are a family of continuous probability distributions defined by the minimum (a), most likely (b) and maximum (c) values that a variable can take. It is a transformation of the four-parameter beta distribution with an additional assumption that its expected value is

The mean of the distribution is therefore defined as the weighted average of the minimum, most likely and maximum values that the variable may take, with four times the weight applied to the most likely value. This assumption about the mean was first proposed in Clark, 1962[1] for estimating the effect of uncertainty of task durations on the outcome of a project schedule being evaluated using the program evaluation and review technique, hence its name. The mathematics of the distribution resulted from the authors' desire to make the standard deviation equal to about 1/6 of the range.[2][3] The PERT distribution is widely used in risk analysis[4] to represent the uncertainty of the value of some quantity where one is relying on subjective estimates, because the three parameters defining the distribution are intuitive to the estimator. The PERT distribution is featured in most simulation software tools.

Comparison with the triangular distribution

[edit]
Comparing density curves for the PERT and triangular probability distributions

The PERT distribution offers an alternative[5] to using the triangular distribution which takes the same three parameters. The PERT distribution has a smoother shape than the triangular distribution. The triangular distribution has a mean equal to the average of the three parameters:

which (unlike PERT) places equal emphasis on the extreme values which are usually less-well known than the most likely value, and is therefore less reliable. The triangular distribution also has an angular shape that does not match the smoother shape that typifies subjective knowledge.

The modified-PERT distribution

[edit]
Comparing density curves for the modified PERT distributions with different weights

The PERT distribution assigns very small probability to extreme values, particularly to the extreme furthest away from the most likely value if the distribution is strongly skewed.[6][7] The Modified PERT distribution [8] was proposed to provide more control on how much probability is assigned to tail values of the distribution. The modified-PERT introduces a fourth parameter that controls the weight of the most likely value in the determination of the mean:

Typically, values of between 2 and 3.5 are used for and have the effect of flattening the density curve; the unmodified PERT would use . This is useful for highly skewed distributions where the distances and are of very different sizes.

The modified-PERT distribution has been implemented in several simulation packages and programming languages:

References

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PERT distribution

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from Grokipedia
The PERT distribution, short for distribution, is a continuous specifically designed for modeling uncertainty in project activity durations within frameworks. It is a specialized form of the , parameterized by three expert-provided estimates: the optimistic (minimum) duration aa, the most likely (mode) duration mm, and the pessimistic (maximum) duration bb. The shape parameters of the underlying are derived as α=1+4m−ab−a\alpha = 1 + 4 \frac{m - a}{b - a} and β=1+4b−mb−a\beta = 1 + 4 \frac{b - m}{b - a}, yielding a duration of μ=a+4m+b6\mu = \frac{a + 4m + b}{6} and a standard deviation of σ=b−a6\sigma = \frac{b - a}{6}. Developed in 1958 as a core component of the original PERT methodology by the Navy's Special Projects Office, in collaboration with consultants from , the distribution was introduced to address scheduling challenges in the Polaris nuclear submarine missile program. This approach marked a shift from deterministic scheduling methods like the (CPM) by incorporating probabilistic elements to better capture real-world variability in task times, enabling managers to compute expected project durations and approximate confidence intervals for completion dates. In practice, the PERT distribution facilitates techniques, where the weighted emphasis on the most likely estimate (via the factor of 4 in the formula) reflects judgment that moderate outcomes are more probable than extremes. It supports both analytical approximations—such as assuming activity times follow independent beta distributions for near-normal project duration distributions—and simulations for more precise assessments in . While the fixed standard deviation assumption simplifies calculations, it has been critiqued for underestimating variance in some scenarios, leading to alternatives like triangular or lognormal distributions in modern tools.

Introduction

Definition and Purpose

The PERT distribution is a specialized continuous probability distribution employed in project management to model uncertainty in activity durations. It is a specific case of the beta distribution, initially defined on the interval [0, 1] and then linearly scaled and shifted to the interval [a, b], where a represents the optimistic (minimum) time estimate and b the pessimistic (maximum) time estimate for a task. This scaling enables the distribution to directly represent real-world time units, such as days or weeks, while preserving the flexible, bounded shape of the beta family for capturing skewed or asymmetric uncertainties. The primary purpose of the PERT distribution is to quantify and incorporate variability in time estimates for individual activities within the (PERT), a network-based project scheduling method. It relies on three-point estimates provided by experts: the optimistic time a, the most likely (modal) time m, and the pessimistic time b. These inputs allow for a probabilistic assessment of task completion times, facilitating analysis, critical path determination, and overall duration forecasting by weighting the most likely estimate more heavily than the extremes. The expected duration is given by the weighted average a+4m+b6\frac{a + 4m + b}{6}, and the variance by (b−a6)2\left(\frac{b - a}{6}\right)^2, assumptions that emphasize while accounting for potential overruns or underruns. In contrast to the general beta distribution, which requires estimating two flexible shape parameters α\alpha and β\beta to fit data, the PERT distribution simplifies this by deriving the shape parameters α=1+4m−ab−a\alpha = 1 + 4 \frac{m - a}{b - a} and β=1+4b−mb−a\beta = 1 + 4 \frac{b - m}{b - a} to match the three-point inputs, yielding α=β=3\alpha = \beta = 3 in the symmetric case (where m=a+b2m = \frac{a + b}{2}) and approximating the specified mean and variance. This parameterization yields a unimodal, bell-shaped that approximates normality under typical estimate spreads, promoting computational ease in manual or early computerized PERT implementations without sacrificing essential modeling fidelity. The forms the foundational family underlying this approach. For instance, in modeling a task with optimistic estimate a=2a = 2 days, most likely m=5m = 5 days, and pessimistic b=10b = 10 days, the PERT distribution assigns higher probability around 5 days but with tails extending to 2 and 10 days, enabling estimation of, say, an 80% chance of completion within 7 days through integration of the scaled —thus aiding managers in buffering schedules against uncertainty.

Historical Background

The (PERT), which incorporates a specialized for modeling activity durations, originated in 1958 as a tool developed by the U.S. Navy's Special Projects Office to manage the complexities of the missile program. This effort involved close collaboration with Lockheed Missile Systems Division, the primary contractor, and , a firm that provided expertise in planning and scheduling. The initiative was driven by the need to handle uncertainty in large-scale projects, where traditional deterministic methods proved inadequate for tracking progress and resources. A key milestone came with the 1959 publication of the seminal paper "Application of a Technique for Program Evaluation" by D.G. Malcolm, J.H. Roseboom, C.E. , and W. Fazar in Operations Research. Willard Fazar, as head of the Navy's techniques development group, played a central role in assembling the interdisciplinary team and guiding the methodology's formulation. The paper introduced the approach—using optimistic, most likely, and pessimistic time estimates—to represent activity durations probabilistically, approximating them with a for computational simplicity in early network analysis. Initially, the PERT distribution employed a classical with shape parameters α = β = 4, which provided a standard deviation roughly one-sixth of the range between optimistic and pessimistic estimates, facilitating variance calculations for project completion times. This choice was later clarified by C.E. Clark in his 1962 letter to the editor in , emphasizing its role in enabling probabilistic network evaluations on limited computing resources of the era. In the 1960s, PERT was integrated with the (CPM), a deterministic technique developed concurrently by and , leading to standardized hybrid approaches for project scheduling across industries. This combination broadened PERT's adoption beyond defense projects. Subsequent evolutions included refinements in , such as , which implemented the PERT distribution's 1-4-1 weighting scheme (derived from the α = β = 4 parameters) to automate three-point estimates and risk analysis.

Mathematical Properties

Probability Density and Cumulative Distribution Functions

The PERT distribution is a continuous supported on the finite interval [a,b][a, b], where aa denotes the optimistic estimate and bb the pessimistic estimate of a project activity duration. It is mathematically equivalent to a four-parameter beta distribution, with the probability density function (PDF) expressed as f(x∣a,b,α,β)=Γ(α+β)Γ(α)Γ(β)(b−a)α+β−1(x−a)α−1(b−x)β−1,a≤x≤b,f(x \mid a, b, \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\beta) (b - a)^{\alpha + \beta - 1}} (x - a)^{\alpha - 1} (b - x)^{\beta - 1}, \quad a \leq x \leq b, and f(x)=0f(x) = 0 otherwise, where α>0\alpha > 0 and β>0\beta > 0 are shape parameters, and Γ\Gamma is the . In the original symmetric formulation of the PERT method, assuming the most likely value mm equals (a+b)/2(a + b)/2, the shape parameters are set to α=4\alpha = 4 and β=4\beta = 4 to match the approximated mean and variance. The (CDF) is the integral of the PDF from aa to xx: F(x∣a,b,α,β)=∫axf(t∣a,b,α,β) dt,a≤x≤b.F(x \mid a, b, \alpha, \beta) = \int_a^x f(t \mid a, b, \alpha, \beta) \, dt, \quad a \leq x \leq b. This can be computed efficiently via the regularized incomplete beta function after a linear transformation y=(x−a)/(b−a)y = (x - a)/(b - a) to the standard beta scale on [0,1][0, 1]: F(x)=Iy(α,β)=1B(α,β)∫0ytα−1(1−t)β−1 dt,F(x) = I_y(\alpha, \beta) = \frac{1}{B(\alpha, \beta)} \int_0^y t^{\alpha - 1} (1 - t)^{\beta - 1} \, dt, where B(α,β)=Γ(α)Γ(β)/Γ(α+β)B(\alpha, \beta) = \Gamma(\alpha) \Gamma(\beta) / \Gamma(\alpha + \beta) is the beta function and Iy(α,β)I_y(\alpha, \beta) is the regularized incomplete beta function. The scaling from the standard beta distribution on [0,1][0, 1] to the practical interval [a,b][a, b] is achieved via the affine transformation x=a+(b−a)yx = a + (b - a) y, where y∼Beta(α,β)y \sim \text{Beta}(\alpha, \beta); this adjustment maps the unit interval to the range of possible activity durations while retaining the flexible, unimodal shape suitable for modeling expert estimates. When parameters α\alpha and β\beta are derived from three-point estimates aa, mm, and bb, the resulting PDF typically produces a smooth, bell-shaped curve centered near the mode mm, with skewness determined by the position of mm relative to [a,b][a, b]; for instance, if mm is closer to aa than to bb, the distribution exhibits positive skew, assigning higher density to lower values while still bounding the support strictly within [a,b][a, b].

Parameter Estimation Methods

The parameter estimation for the PERT distribution, a special case of the scaled on the interval [a, b], relies on three expert-provided point estimates: the optimistic (minimum) value a, the most likely (mode) value m, and the pessimistic (maximum) value b, where a < m < b. The shape parameters α and β are conventionally estimated using a heuristic that weights the mode m four times more heavily than the extremes a and b, reflecting the belief that the most likely outcome dominates uncertainty in project durations. This approach stems from the original PERT methodology's assumptions about activity times following a , with parameters fitted via approximate quantiles. The standard formulas for the shape parameters are: α=1+4(m−a)b−a\alpha = 1 + \frac{4(m - a)}{b - a} β=1+4(b−m)b−a\beta = 1 + \frac{4(b - m)}{b - a} These ensure that the resulting distribution has mean μ=a+4m+b6\mu = \frac{a + 4m + b}{6} and approximates the variance as σ2=(b−a)236\sigma^2 = \frac{(b - a)^2}{36}, aligning with traditional PERT approximations while fixing α + β = 6 for simplicity. The mode m corresponds exactly to the distribution's peak for the beta form, with the distribution symmetric (α = β = 3) when m = (a + b)/2. This equal weighting of extremes leads to the symmetric case, emphasizing central tendency when expert estimates are balanced. To illustrate, consider estimates a = 2, m = 5, b = 10 for an activity duration. The range b - a = 8. Then α = 1 + 4(5 - 2)/8 = 1 + 12/8 = 2.5 and β = 1 + 4(10 - 5)/8 = 1 + 20/8 = 3.5.

Statistical Moments and Characteristics

The , parameterized by the minimum value aa, mode mm, and maximum value bb, derives its statistical moments from its formulation as a scaled and shifted with shape parameters α=1+4m−ab−a\alpha = 1 + 4 \frac{m - a}{b - a} and β=1+4b−mb−a\beta = 1 + 4 \frac{b - m}{b - a}. These parameters ensure α+β=6\alpha + \beta = 6, emphasizing the mode through a weighting factor of four. The mean μ\mu is exactly μ=a+(b−a)αα+β=a+4m+b6,\mu = a + (b - a) \frac{\alpha}{\alpha + \beta} = \frac{a + 4m + b}{6}, providing a weighted average that places greater emphasis on the most likely outcome mm. This exact expression aligns with the standard PERT formula originally proposed for project duration estimates. The variance σ2\sigma^2 is given exactly by the beta distribution formula σ2=(b−a)2αβ(α+β)2(α+β+1)=(b−a)2αβ252,\sigma^2 = (b - a)^2 \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} = \frac{(b - a)^2 \alpha \beta}{252}, where αβ\alpha \beta varies with the mode's position, typically ranging from about 4 to 25 for mm between aa and bb. The conventional PERT approximation simplifies this to σ2≈(b−a6)2=(b−a)236,\sigma^2 \approx \left( \frac{b - a}{6} \right)^2 = \frac{(b - a)^2}{36}, assuming the range b−ab - a spans six standard deviations, akin to a normal distribution's near-full coverage. This approximation slightly overestimates the true variance when the mode is not centered but remains widely used for its computational ease in network analysis. The PERT distribution is generally asymmetric, exhibiting positive when mm is closer to aa (i.e., β>α\beta > \alpha) and negative when closer to bb, with zero only in the symmetric case m=(a+b)/2m = (a + b)/2. The coefficient follows the beta distribution's form γ1=2(β−α)α+β+1(α+β+2)αβ,\gamma_1 = \frac{2 (\beta - \alpha) \sqrt{\alpha + \beta + 1}}{(\alpha + \beta + 2) \sqrt{\alpha \beta}},
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