# Random Variable

**Type:** Concept  
**Domain:** Probability & Statistics  
**Codex URL:** /mathematics/probability-statistics/random-variable/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A function assigning numerical values to outcomes of a random process. Formally X:Ω→ℝ, measurable.

## Discrete vs continuous
Discrete: distinct values (die roll). Continuous: any value in a range (height, time).

## Expected value
E[X] = Σx·p(x) (discrete) or ∫x·f(x)dx (continuous). Fair die: E[X]=3.5.

## Variance & standard deviation
Var(X)=E[(X−E[X])²]; σ=√Var(X)

## Linearity of expectation
E[X+Y]=E[X]+E[Y] — holds even without independence.

## Measure-theoretic definition
X:(Ω,F,P)→(ℝ,B), a measurable function. E[X]=∫_Ω X dP (Lebesgue integral).

## Convergence types
Convergence in distribution (weakest, CLT), in probability (weak LLN), almost sure (strongest, strong LLN).

## Sources
### Tier 1
- Feller, W. (1968/1971). *An Introduction to Probability Theory*. Vols 1-2, 3rd ed.
- Billingsley, P. (1995). *Probability and Measure*. 3rd ed.
### Tier 2
- Ross, S.M. (2014). *A First Course in Probability*. 9th ed.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*