Distributions

Introduction

The package includes a collection of probabilistic distributions and related functions. The implementation is heavily influenced by the package Distributions.jl, but extends it to allow for weighted distributions as well as the Unitful.jl parameters.

Distributions

Each distribution has a number of constructors, both based on its full name and abbreviation, e.g., Weibull() is equivalent to 𝑾(). The abbreviation constructor is the first letter of the distribution in the \bi font. Furthermore, additional constructors are included for limited input, where all remaining parameters are set to their default, e.g., Weibull(10.0u"hr") is equivalent to Weibull(10.0u"hr",1.0,1.0).

All input parameters are either of the type Number, its subtype Real or a Function:

Number
 |- Unitful.Quantity, e.g., 10.0u"hr"
 |- Real
     |- AbstractFloat, e.g., 10.0
     |- Integer, e.g., 10
Function

All distributions may be scaled using a weight parameter ω, where 0.0 < ω::Real ≤ 1.0 or 0.0 < ω(t)::Fuction ≤ 1.0.

Exponential Distribution

The exponential distribution with scale parameter θ and an optional weight ω has a probability density function

\[f(x; θ, ω) = \begin{cases} ω/θ e^{-x/θ} & x ≥ 0, \\ 0 & x < 0. \end{cases}\]

Weibull Distribution

The Weibull distribution with scale parameter θ, shape parameter α and optional weight ω has a probability density function

\[f(x, θ, α, ω) = \begin{cases} \frac{αω}{θ} \cdot \left(\frac{x}{θ}\right)^{α-1} \cdot e^{-\left(\frac{x}{θ}\right)^{α}} &\text{if } x ≥ 0, \\ 0 &\text{if } x < 0. \end{cases}\]

Constructors

Examples

julia> Weibull()            # default Weibull distr. with θ = 1.0, α = 1.0 and ω = 1.0
julia> 𝑾(3.0u"minute")     # Weibull distr. with θ = 3.0 min, α = 1.0 and ω = 1.0
julia> 𝑾(5.0u"yr",4.0)     # Weibull distr. with θ = 5.0 yr, α = 4.0 and ω = 1.0
julia> 𝑾(10.0,0.5,0.2)     # scaled Weibull distr. with θ = 10.0, α = 0.5 and ω = 0.2

Normal Distribution

The Normal distribution (also called Gaussian distribution) with mean μ, standard deviation σ and optional weight ω has a probability density function

\[f(x, μ, σ, ω) = \frac{ω}{\sqrt{2π} σ} \cdot e^{-\frac{(x-μ)^{2}}{2 σ^{2}}}\]

The normal distribution is widely used in reliability engineering for modeling measurement errors, natural variations, and as a limiting distribution in many statistical applications. In the LVDC examples it is mainly used to model the uncertainty around the measurements and disconnection times of the circuit breakers.

Constructors

Examples

julia> Normal()             # standard normal distr. with μ = 0.0, σ = 1.0 and ω = 1.0
julia> 𝑵(100.0u"hour")     # normal distr. with μ = 100.0 hr, σ = 1.0 hr and ω = 1.0
julia> 𝑵(50.0u"MW", 5.0u"MW")  # normal distr. with μ = 50.0 MW, σ = 5.0 MW and ω = 1.0
julia> 𝑵(25.0, 3.0, 0.95)  # scaled normal distr. with μ = 25.0, σ = 3.0 and ω = 0.95

LogNormal Distribution

The Log-normal distribution with expected value μ and standard deviation σ of the corresponding normal distribution and optional weight ω has a probability density function

\[f(x, μ, σ, ω) = \begin{cases} \frac{ω}{\sqrt{2π} x σ} \cdot e^{-\frac{(\ln{x}-μ)^{2}}{2 σ^{2}}} &\text{if } x ≥ 0, \\ 0 &\text{if } x < 0. \end{cases}\]

Given the ln-function, all Unitful values are converted to correspond with the unit of μ.

Constructors

Examples

julia> LogNormal()          # default Log-normal distr. with μ = 1.0, σ = 1.0 and ω = 1.0
julia> 𝑳(3.0u"minute")      # Log-normal distr. with μ = 3.0 min, σ = 1.0 min and ω = 1.0
julia> 𝑳(5.0u"yr",4.0u"d")  # Log-normal distr. with μ = 5.0 yr, σ = 4.0 d and ω = 1.0
julia> 𝑳(10.0,0.5,0.2)      # scaled Log-normal distr. with μ = 10.0, σ = 0.5 and ω = 0.2