General random variable

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

𝑋 : 𝜉 \to X

and the probability of any measurable $𝐴 \in Σ$ is given by the induced measure $ℙ \circ 𝑋 - 1 ↾ Σ$ , i.e.

ℙ [𝑋 \in 𝐴] = ℙ {𝜖 \in 𝜉 : 𝑋 (𝜖 \in 𝐴)}

#state/tidy | #lang/en | #SemBr