Probability Theory 10 | Random Variables [dark version]

Random variables turn the outcomes of a random experiment into a single, well-defined numerical object—by requiring that the mapping from outcomes to numbers respects the event structure of probability theory. Concretely, a random variable is a function X from a sample space Ω into another set Ω̃ (often the real numbers), but it only counts as a random variable if every “measurable” event in Ω̃ pulls back to a measurable event in Ω. This measurability condition is what makes probabilities like P(X ∈ Ã) mathematically legitimate.

The discussion starts with a familiar experiment: rolling two distinguishable dice (a red one and a green one). The sample space is the Cartesian product {1,…,6}×{1,…,6}, with the σ-algebra taken as the power set and probabilities given by the uniform distribution. If the game only cares about the sum of the two dice, the relevant object is the random variable X defined by X(ω1, ω2)=ω1+ω2. Here, the input is an outcome pair from Ω, and the output is a number—exactly the kind of “information extraction” random variables are meant to provide.

To formalize the idea, the transcript gives the general definition using measurable (event) spaces. One starts with a measurable space (Ω, A) and another (Ω̃, Ã). A map X:Ω→Ω̃ is called a random variable if it is measurable in the measure-theoretic sense: for every event à in Ã, the pre-image X^{-1}(Ã) must lie in A. That requirement ensures that once a probability measure P is fixed on (Ω, A), the probability of events described in terms of X can be computed.

Two examples show why the condition matters. In the dice-sum setup, the σ-algebra on Ω is the full power set, so any pre-image of a set in Ω̃ automatically lands in A. That makes X a random variable without any real work. But if the σ-algebra on Ω is shrunk to the smallest possible one—{∅, Ω}—then measurability can fail. For instance, consider the event “the sum equals 2.” Its pre-image is the set of outcomes where both dice show 1, which is neither empty nor all of Ω, so it is not in the reduced σ-algebra. In that case, X is not a random variable.

The closing notation ties measurability to probability calculations. Once X is a random variable and P is defined on (Ω, A), probabilities of the form P(X ∈ Ã) are well-defined because X^{-1}(Ã) belongs to A. The transcript also introduces shorthand notation like P(X ∈ Ã) written as P(X ∈ Ã) or even P(X ∈ ã)-style expressions, emphasizing that the left-hand side is a compact way to refer to the probability of the corresponding pre-image set in Ω. The key takeaway: random variables are not just functions to numbers—they are functions whose event structure matches the σ-algebras so probabilities can be assigned consistently.