Continuous Distributions
UniDist.jl provides 54 continuous probability distributions. This page documents each distribution with its parameters, support, and usage examples.
Common Distributions
Normal (Gaussian)
The most widely used continuous distribution, characterized by its bell-shaped curve.
Normal(μ, σ) # μ = mean, σ = standard deviationParameters:
μ(mu): Mean (location parameter)σ(sigma): Standard deviation (scale parameter), σ > 0
Support: (-∞, +∞)
Example:
d = Normal(0.0, 1.0) # Standard normal
pdf(d, 0.0) # ≈ 0.3989
cdf(d, 1.96) # ≈ 0.975
quantile(d, 0.975) # ≈ 1.96
mean(d) # 0.0
var(d) # 1.0Use cases: Modeling measurement errors, natural phenomena, test scores, financial returns.
Exponential
Models the time between events in a Poisson process.
Exponential(θ) # θ = scale parameter (mean)Parameters:
θ(theta): Scale parameter (mean), θ > 0
Support: [0, +∞)
Example:
d = Exponential(2.0) # Mean time = 2
pdf(d, 1.0) # ≈ 0.303
cdf(d, 2.0) # ≈ 0.632
mean(d) # 2.0Use cases: Waiting times, survival analysis, reliability engineering, radioactive decay.
Uniform
All values in an interval are equally likely.
Uniform(a, b) # a = lower bound, b = upper boundParameters:
a: Lower boundb: Upper bound, b > a
Support: [a, b]
Example:
d = Uniform(0.0, 10.0)
pdf(d, 5.0) # 0.1
cdf(d, 5.0) # 0.5
quantile(d, 0.5) # 5.0
mean(d) # 5.0Use cases: Random number generation, modeling complete uncertainty within bounds.
Beta
Flexible distribution on [0, 1], often used for probabilities.
Beta(α, β) # α, β = shape parametersParameters:
α(alpha): Shape parameter, α > 0β(beta): Shape parameter, β > 0
Support: [0, 1]
Example:
d = Beta(2.0, 5.0)
pdf(d, 0.3) # ≈ 2.06
cdf(d, 0.5) # ≈ 0.89
mean(d) # ≈ 0.286Use cases: Bayesian inference for proportions, modeling probabilities, A/B testing.
Gamma
Generalization of exponential distribution, models waiting times for multiple events.
Gamma(α, θ) # α = shape, θ = scaleParameters:
α(alpha): Shape parameter, α > 0θ(theta): Scale parameter, θ > 0
Support: [0, +∞)
Example:
d = Gamma(2.0, 2.0)
pdf(d, 2.0) # ≈ 0.184
cdf(d, 4.0) # ≈ 0.594
mean(d) # 4.0 (α * θ)Use cases: Waiting times, insurance claims, rainfall amounts, Bayesian inference.
Location-Scale Distributions
Cauchy
Heavy-tailed distribution with no defined mean or variance.
Cauchy(x₀, γ) # x₀ = location, γ = scaleParameters:
x₀: Location parameterγ(gamma): Scale parameter, γ > 0
Support: (-∞, +∞)
Example:
d = Cauchy(0.0, 1.0) # Standard Cauchy
pdf(d, 0.0) # ≈ 0.318
cdf(d, 0.0) # 0.5Use cases: Modeling outliers, ratio of normal variables, resonance in physics.
Laplace (Double Exponential)
Symmetric distribution with heavier tails than normal.
Laplace(μ, b) # μ = location, b = scaleParameters:
μ(mu): Location parameterb: Scale parameter, b > 0
Support: (-∞, +∞)
Example:
d = Laplace(0.0, 1.0)
pdf(d, 0.0) # 0.5
cdf(d, 0.0) # 0.5Use cases: Finance, signal processing, LASSO regression, robust estimation.
Logistic
Similar to normal but with heavier tails.
Logistic(μ, s) # μ = location, s = scaleParameters:
μ(mu): Location (mean)s: Scale parameter, s > 0
Support: (-∞, +∞)
Example:
d = Logistic(0.0, 1.0)
pdf(d, 0.0) # 0.25
cdf(d, 0.0) # 0.5Use cases: Logistic regression, growth models, neural networks.
Chi-Square Family
ChiSquare
Sum of squared standard normal variables.
ChiSquare(ν) # ν = degrees of freedomParameters:
ν(nu): Degrees of freedom, ν > 0
Support: [0, +∞)
Example:
d = ChiSquare(5)
pdf(d, 3.0) # ≈ 0.154
cdf(d, 5.0) # ≈ 0.584
mean(d) # 5.0Use cases: Hypothesis testing, confidence intervals, goodness-of-fit tests.
Chi
Square root of chi-square distribution.
Chi(ν) # ν = degrees of freedomParameters:
ν(nu): Degrees of freedom, ν > 0
Support: [0, +∞)
F
Ratio of two chi-square distributions.
F(ν₁, ν₂) # ν₁, ν₂ = degrees of freedomParameters:
ν₁: Numerator degrees of freedomν₂: Denominator degrees of freedom
Support: [0, +∞)
Example:
d = F(5, 10)
pdf(d, 1.0) # ≈ 0.61
cdf(d, 2.0) # ≈ 0.84Use cases: ANOVA, comparing variances, regression analysis.
Extreme Value Distributions
ExtremeValue (Gumbel)
Models the maximum of many samples.
ExtremeValue(μ, σ) # μ = location, σ = scaleParameters:
μ(mu): Location parameterσ(sigma): Scale parameter, σ > 0
Support: (-∞, +∞)
Use cases: Extreme weather events, flood analysis, material strength.
Weibull
Flexible distribution for reliability and survival analysis.
Weibull(α, θ) # α = shape, θ = scaleParameters:
α(alpha): Shape parameter, α > 0θ(theta): Scale parameter, θ > 0
Support: [0, +∞)
Example:
d = Weibull(2.0, 1.0)
pdf(d, 0.5) # ≈ 0.779
cdf(d, 1.0) # ≈ 0.632Use cases: Reliability engineering, failure analysis, wind speed modeling.
Pareto
Heavy-tailed distribution following power law.
Pareto(α, θ) # α = shape, θ = scale (minimum)Parameters:
α(alpha): Shape parameter, α > 0θ(theta): Scale (minimum value), θ > 0
Support: [θ, +∞)
Use cases: Income distribution, city populations, file sizes, insurance claims.
Log-Transformed Distributions
LogNormal
Distribution of exp(X) where X is normal.
LogNormal(μ, σ) # μ, σ = parameters of log(X)Parameters:
μ(mu): Mean of log(X)σ(sigma): Standard deviation of log(X), σ > 0
Support: (0, +∞)
Example:
d = LogNormal(0.0, 1.0)
pdf(d, 1.0) # ≈ 0.399
cdf(d, 1.0) # 0.5Use cases: Stock prices, income, biological measurements, particle sizes.
LogLogistic
Log-transformed logistic distribution.
LogLogistic(α, β) # α = scale, β = shapeSupport: (0, +∞)
Use cases: Survival analysis, hydrology, economics.
Other Continuous Distributions
Arcsin
U-shaped distribution on [0, 1].
Arcsin(a, b) # a = lower bound, b = upper boundSupport: [a, b]
Arctangent
Arctangent(θ, σ)Erlang
Special case of Gamma with integer shape.
Erlang(k, θ) # k = shape (integer), θ = scaleUse cases: Queuing theory, telecommunications.
Error
Error(μ, σ, p)ExponentialPower (Generalized Normal)
ExponentialPower(μ, σ, p)GeneralizedGamma
GeneralizedGamma(a, d, p)GeneralizedPareto
GeneralizedPareto(μ, σ, ξ)Use cases: Extreme value analysis, tail risk modeling.
Gompertz
Gompertz(η, b)Use cases: Mortality modeling, actuarial science.
HyperbolicSecant
HyperbolicSecant(μ, σ)Hyperexponential
Hyperexponential(probs, rates)Hypoexponential
Hypoexponential(rates)IDB (Inverse Beta Distribution)
IDB(α, β, θ)InverseGaussian
InverseGaussian(μ, λ)Use cases: First passage times, reliability.
InvertedBeta
InvertedBeta(α, β)InvertedGamma
InvertedGamma(α, θ)Use cases: Bayesian inference for variance.
KolmogorovSmirnov
KolmogorovSmirnov()Use cases: Goodness-of-fit testing.
LogGamma
LogGamma(α, β)LogisticExponential
LogisticExponential(λ, κ)Lomax (Pareto Type II)
Lomax(α, λ)Makeham
Makeham(a, b, c)Use cases: Mortality modeling.
Minimax
Minimax(β, γ)Muth
Muth(α)Power
Power(α, a, b)Rayleigh
Rayleigh(σ)Use cases: Wind speed, wave height, signal processing.
Triangular
Triangular(a, b, c) # a = min, b = max, c = modeUse cases: Project management (PERT), when only min/max/mode are known.
VonMises
Circular distribution for angular data.
VonMises(μ, κ) # μ = mean direction, κ = concentrationUse cases: Wind directions, compass bearings, time-of-day data.
Noncentral Distributions
NoncentralBeta
NoncentralBeta(α, β, λ)NoncentralChiSquare
NoncentralChiSquare(ν, λ)Use cases: Power analysis, signal detection.
NoncentralF
NoncentralF(ν₁, ν₂, λ)NoncentralT
NoncentralT(ν, λ)Use cases: Power analysis, effect size calculations.
DoublyNoncentralF
DoublyNoncentralF(ν₁, ν₂, λ₁, λ₂)DoublyNoncentralT
DoublyNoncentralT(ν, λ₁, λ₂)Standard (Parameterless) Distributions
These distributions have fixed parameters for convenience:
StandardNormal() # Normal(0, 1)
StandardUniform() # Uniform(0, 1)
StandardCauchy() # Cauchy(0, 1)
StandardTriangular() # Triangular(-1, 1, 0)
StandardPower(α) # Power distribution on [0, 1]Example:
d = StandardNormal()
pdf(d, 0.0) # ≈ 0.3989
cdf(d, 1.96) # ≈ 0.975