Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Real random variables on finite probability spaces and their finite distributions
Definition
A real random variable on a finite probability space is a function . Its range is finite. The distribution or law of is the finite probability space on with The weights are nonnegative and sum to because the fibres of form a finite partition of (Probability is additive on every finite pairwise-disjoint family of events).
Depends on
- Finite probability spaces, outcome weights, events, and event probabilities
- A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$; $f : A \to B$, the value $f(a)$, domain and codomain
- Finite sums and finite products, by recursion
- Probability is additive on every finite pairwise-disjoint family of events
Used by
- Markov's conclusion can fail without nonnegativity Counterexample
- Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials Definition
- Expectation of a real random variable on a finite probability space Definition
- Pairwise and mutual independence of finite-valued random variables Definition
- The indicator random variable of an event Definition
- A symmetric two-point distribution attains equality in Chebyshev's inequality Example
- A two-valued random variable attains equality in Markov's inequality Example
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 57 results over 23 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Matousek and J. Vondrak, The Probabilistic Method, Definition 1.1.6 (standard reference, not scraped)
- M. Bucic, Probabilistic Method, Definition A.4 (standard reference, not scraped)