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.
The expectation of an indicator is the probability of the event
Statement
Let be a probability space and let . Then the indicator satisfies
Facts & Assumptions
Given: A probability space and an event .
Expectation of a nonnegative random variable is its integral with respect to (Expectation of a nonnegative or integrable random variable).
The complement identity gives (Basic identities for a probability measure).
On nonnegative simple functions, the nonnegative integral is the simple integral (The nonnegative integral agrees with the simple integral on simple functions, The integral of a nonnegative simple function).
Proof
The function is the simple function Therefore [L1] and [L3] give
Using [L2] only to note that is the complementary event in the same probability space, step 1.1 simplifies to .
Depends on
Used by
- The second-moment lower bound for positive probability Corollary
- A measurable [0,1]-valued statistic need not be a valid p-value Counterexample
- Critical, detection, and quantification levels Definition
- A continuous likelihood-ratio test Example
- Strong law for empirical indicator averages 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
- A PRDS bound for each BH true-null contribution Lemma
- A random inner linear code has fewer than one bad word in expectation Lemma
- Finite-horizon crossing bound for a nonnegative test supermartingale Lemma
- Selected-size conditional validity yields a conditionally superuniform p-value Lemma
- Benjamini–Hochberg FDR under independence Theorem
- BY controls FDR under arbitrary dependence Theorem
- Conditional-class standardization identifies a causal mean Theorem
- Conditional-error budget for two-stage adaptation Theorem
- Conformal rank superuniformity under exchangeability Theorem
- Exact limits in the known-blank equal-variance normal model Theorem
- Horvitz–Thompson variance and pair-inclusion estimate Theorem
- Inverse probability weighting identifies a causal mean Theorem
- Pairwise-independent Borel-Cantelli frequency law Theorem
- Sen–Yates–Grundy variance identity Theorem
- Valid p-values yield level tests for nonempty disjoint alternatives Theorem
Dependency tree · two levels
18 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
- Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.6 (standard reference, not scraped)