RandomVariables.jl

A Julia package for random variables, some transformations and probabilities
Author ManuelStapper
Popularity
15 Stars
Updated Last
11 Months Ago
Started In
June 2022

RandomVariables

Build status Coverage DOI

A Julia package for random variables and probabilities as an add-on to the Distributions.jl package. Scope of the package:

  • Defining a random variable X = RV(Normal(0, 1)); Y = RV(Poisson(4))
  • Transformation of random variables: Z = log(Y + 1)
  • Convenient probability computation: P(X < 2)
  • Combination of events: A = (X < 2) & (Y ≥ 2); B = (X ≤ 5) & (Y > 0)
  • Defining conditional events: P(A|B)

Installation

Distributions.jl package must be installed and loaded to use RandomVariables.jl

julia> import Pkg
julia> Pkg.add("Distributions")
julia> Pkg.add("RandomVariables")
julia> using RandomVariables, Distributions

Random variables

To define a random variable, the function RV() takes any univariate distribution from the Distributions.jl package and equips it with an ID.

X = RV(Normal())
Y = RV(Poisson(4))

Transformation of random variables

Random variables can be transformed. Say you transform a random variable X by a function h(x), i.e. Z = f(X). The object RVtransformed stores the distribution of the original random variable X, its ID, the function h and the inverse of the transformation function. Implemented transformations for a real valued y are: X + y, X - y, X * y, X / y, inv(X), abs(X), exp(X), log(X), sqrt(X) and X^y.

Events

To compute probabilities, events can be defined as a combination of independent random variables and bounding intervals. For example X > 1 creates an event with random variable X and the interval (1, ∞). The bounding intervals can be closed [a, b], open (a, b) or semi-closed [a, b) or (a, b]. Two events can be combined with operators

  • & : both events occur
  • : Either one event occurs or both
  • \ : Left event occurs but not the right event
  • : One event occurs but not the other
  • ! : The complementary event

Examples for events:

A = X > 1
B = X ≤ 3
C = X ≤ 0
  • A & B : (1, 3]
  • A ∨ C : (-∞, 0] ∪ (1, ∞)
  • A \ B : (3, ∞)
  • A ⊻ B : (-∞, 1] ∪ (3, ∞)
  • !(A) : (-∞, 1]

Events can include multiple independent random variables. For n variables, the bounding intervals will then be n dimensional rectangles (boxes). For example (X < 1) & (X ≥ 0) & (Y ≤ 1) & (Y > 0) creates an event with random variables X and Y with bounding rectangle [0, 1) × (0, 1].

Conditional Events

With two events, conditional events can be defined. A|B gives the event that event A occurs given that B occurs. The random variables in A do not necessarily need to match the random variables in B.

Example:

A = (X > 1) & (log(Y + 1) ≤ 1)
B = (X < 5)
P(A|B)

Moments

Mean, variance, skewness and kurtosis functions from the Distributions.jl package can also be applied to random variables. Further, a short notation for the mean E(X) is added. The mean of a transformed random variable is computed by numerical integration / summation.

Example

P((X - E(X))^2 < 3)
E(log(Y + 1))

Outlook

Short term

  • A more detailed documentation
  • Conditional moments, i.e. E(X | X > 1)
  • Shift to Intervals.jl (?)
  • Minimum and maximum, i.e. Z = minimum(X, Y)
  • Transfer more functionalities of Distributions.jl to random variables?
  • Plotting functions?

Long term

  • More arithmetic on random variables: X + Y, ...
  • Inclusion of dependent random variables in general

References

Distributions.jl package

Used By Packages

No packages found.