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.
Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials
Definition
For , a Bernoulli random variable takes the value with probability and with probability .
For , a binomial random variable is a sum of mutually independent Bernoulli variables. When , this is the constant zero random variable.
Depends on
Used by
- De Moivre-Laplace central limit theorem Corollary
- The Erdős-Rényi finite random graph G(n,p) Definition
- A Lindeberg array with no identically distributed row Example
- Bernoulli sample frequencies Example
- First- and second-moment bounds for a nonempty Bernoulli random subset Example
- Normal approximation to binomial probabilities Example
- A Bernoulli(p) variable has mean p and variance p(1-p); a binomial(n,p) variable has mean np and variance np(1-p) Lemma
- Chernoff bound for independent bernoulli trials Lemma
Dependency tree · two levels
16 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
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 5.1 (standard reference, not scraped)
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 3.1.5 (standard reference, not scraped)