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.
Pairwise and mutual independence of finite-valued random variables
Definition
A finite family of finite-valued random variables is mutually independent when for every nonempty and every selection , It is pairwise independent when this identity is required only for two distinct indices. The empty family is mutually independent vacuously, and a one-member family is mutually independent.
Depends on
Used by
- The general rectangle criterion agrees with the published finite random-variable definition Corollary
- For dependent variables, E[XY] need not equal E[X]E[Y] Counterexample
- Uncorrelated finite random variables need not be independent Counterexample
- Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials Definition
- Pairwise independent hash family interface Definition
- Probabilistic polynomial time machine interface Definition
- First- and second-moment bounds for a nonempty Bernoulli random subset Example
- False: linearity of expectation requires independence False statement
- A Chernoff bound for sums of independent Bernoulli trials Lemma
- The moment generating function of a finite sum of independent variables is the product of their moment generating functions Lemma
- A finite family of events is mutually independent exactly when its indicators are mutually independent Theorem
- Expectation factors over a finite product of mutually independent random variables Theorem
- For n≥1 independent random signs, ℙ(|S|≥ t)≤2 exp(-t²/(2n)) for t>0 Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Matousek and J. Vondrak, The Probabilistic Method, Definition 1.1.8 (standard reference, not scraped)
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 4.1 (standard reference, not scraped)