Finite discrete probability distribution

Mathematical statement

The probability mass function p of a finite discrete random variable assigns a nonnegative probability to each element of its finite value range, and all masses sum to 1. Event probabilities add the corresponding masses; expectation is the mass-weighted sum of values.

Representative formula (exact TeX)

p:X\to R

Assumptions

• Outcome values are distinct and masses are exact rationals. A qualified PMF requires every mass nonnegative and the total exactly 1.
• Exercises represent mathematical distributions only, not sample estimates, data fits or real populations.

Symbol scope

• p — probability mass function
• X — finite value range
• R — real codomain

Intuition, definition, and misconceptions

Check nonnegativity and total one before using a PMF. An unqualified mass list cannot then be treated as a probability distribution for expectation.

An event probability adds the masses of values in the event; values outside the event are excluded.

A pushforward distribution merges the original masses that map to the same new value. This remains an exact mathematical transformation, not statistical estimation.

Do not ignore total mass merely because individual entries lie between 0 and 1, or present a theoretical PMF as a fitted data model.

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

• Real random variables on finite sample spaces — A distribution describes the probability law of a random variable’s values.
• Probability on finite sample spaces — Masses obey nonnegativity and total one.
• Function mapping — The PMF assigns exactly one mass to each value.

Verification boundary

Only finite exact PMFs are handled; continuous densities, estimation quality and empirical fit are not claimed.

not_formalized

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

Next obligation

Add joint distributions, discrete stochastic processes and finite Markov chains.

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.