Struct Pert 

pub struct Pert { /* private fields */ }

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

impl Pert

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].

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.

pub fn min(&self) -> f64

pub fn mode(&self) -> f64

pub fn max(&self) -> f64

pub fn alpha(&self) -> f64

pub fn beta_param(&self) -> f64

pub fn mean(&self) -> f64

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

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

pub fn variance(&self) -> f64

Exact variance using Beta distribution formula.

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

pub fn std_dev(&self) -> f64

Standard deviation = √(variance).

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).

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

Approximate quantile using normal approximation.

Uses μ + σ·Φ⁻¹(p) with the exact PERT mean and std dev. This is an approximation; accuracy decreases for highly skewed PERTs.

Returns None if p is outside (0, 1).

Trait Implementations

impl Clone for Pert

fn clone(&self) -> Pert

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for Pert

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl PartialEq for Pert

fn eq(&self, other: &Pert) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl StructuralPartialEq for Pert