RandomVariables
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