Probability theory MOC

General random variable

A random variable assigns elements of some measurable space (X,Σ) to possible outcomes/world-states in a probability model (𝜉,F,ℙ). prob Formally a random variable on (𝜉,F,ℙ) of (X,Σ) is a measurable function

𝑋:𝜉→X

and the probability of any measurable 𝐴 ∈Σ is given by the induced measure ℙ ∘𝑋−1 ↾Σ, i.e.

ℙ[𝑋∈𝐴]=ℙ{𝜖∈𝜉:𝑋(𝜖∈𝐴)}

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