Struct Pert

Source

pub struct Pert { /* private fields */ }

Expand description

PERT distribution (Program Evaluation and Review Technique).

A modified Beta distribution defined by three points: optimistic (min), most likely (mode), and pessimistic (max).

§Mathematical Definition

Shape parameters (with λ = 4):

α = 1 + λ · (mode − min) / (max − min)
β = 1 + λ · (max − mode) / (max − min)

The underlying variable Y = (X − min)/(max − min) follows Beta(α, β).

Mean: (min + λ·mode + max) / (λ + 2) = (min + 4·mode + max) / 6
Std Dev (simplified): (max − min) / (λ + 2) = (max − min) / 6

§Exact vs Simplified Variance

The simplified variance ((max−min)/6)² is an approximation. The exact variance uses the Beta distribution formula:

Var = α·β / ((α+β)²·(α+β+1)) × (max−min)²

The simplified formula is exact when the distribution is symmetric (mode = midpoint) and becomes less accurate as skewness increases.

Reference: Malcolm et al. (1959), “Application of a Technique for Research and Development Program Evaluation”, Operations Research 7(5).

Implementations§

Source §

impl Pert

Source

pub fn new(min: f64, mode: f64, max: f64) -> Result<Self, DistributionError>

Creates a standard PERT distribution (λ = 4).

§Errors

Returns Err if min >= max or mode is outside [min, max].

Source

pub fn with_shape( min: f64, mode: f64, max: f64, lambda: f64, ) -> Result<Self, DistributionError>

Creates a modified PERT distribution with custom shape parameter λ.

λ controls the weight of the mode:

λ = 4: standard PERT
λ > 4: tighter distribution (more peaked)
λ < 4: flatter distribution (less peaked)

§Errors

Returns Err if parameters are invalid.

Source

pub fn mean(&self) -> f64

Mean = (min + 4·mode + max) / 6 for standard PERT (λ=4).

General: (min + λ·mode + max) / (λ + 2).

Source

pub fn variance(&self) -> f64

Exact variance using Beta distribution formula.

Var = α·β / ((α+β)²·(α+β+1)) × (max−min)²

Source

pub fn std_dev(&self) -> f64

Standard deviation = √(variance).

Source

pub fn cdf(&self, x: f64) -> f64

CDF via regularized incomplete beta function approximation.

Uses a numerical approximation of the regularized incomplete beta function I_x(α, β).

Source

pub fn quantile(&self, p: f64) -> Option<f64>

Evaluates the quantile (inverse CDF) at probability p ∈ [0, 1].

A PERT is a Beta(α, β) stretched over [min, max], so its quantile is min + (max − min)·B⁻¹(p; α, β) – exact to the tolerance of the Beta quantile, and inside [min, max] by construction. (A normal approximation μ + σ·Φ⁻¹(p) is off for any skewed estimate and can land outside the support.)

Returns None if p is outside [0, 1].

§Examples

use u_numflow::distributions::Pert;
let pert = Pert::new(0.0, 1.0, 10.0).unwrap();
let q = pert.quantile(0.9).unwrap();
assert!((pert.cdf(q) - 0.9).abs() < 1e-9);