Real random variables on finite sample spaces

Mathematical statement

On a nonempty finite sample space S with event family F=𝒫(S) and a valid probability law P, a real random variable is a total function X:S→R. For any real subset T, the preimage {s∈S:X(s)∈T} is an event. P(X=t) sums the weights of all outcomes with X(s)=t.

Representative formula (exact TeX)

X:S\to R

Assumptions

• S is finite and nonempty, F is the full powerset, and X assigns one real value to every elementary outcome.
• Value, range, and preimage-event questions read only the function data without guessing the probability law; point-probability questions separately specify valid rational weights.

Symbol scope

• X — real-valued random variable
• S — nonempty finite sample space
• R — real codomain

Intuition, definition, and misconceptions

Different elementary outcomes may have the same random-variable value. The range removes duplicate values, but point probabilities sum the weights of every preimage.

An explicit condition such as X(s)≥t selects elementary outcomes for a preimage event. The event set is distinct from a numerical probability; missing weights do not imply equiprobability.

Exercises use exact integer outcome labels and integer output values. A finite full powerset makes every preimage an event. General measurable spaces, continuous densities, and distribution representations have separate obligations.

Common errors include identifying a random variable with one sampled outcome, treating range cardinality as probability, or omitting weights of repeated output values.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Function mapping — A random variable is total on its sample space.
• Sample space — Inputs are elementary outcomes rather than random-variable values.
• Real number — Outputs are real numbers.
• Probability on finite sample spaces — Point probabilities require weights and event probabilities.

Verification boundary

Finite functions and point masses can be recomputed completely; general measurability and distribution-pushforward bridges have no attached Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind finite PMFs, total functions, preimage events, and pushforwards; list general Measurable obligations separately.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.