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.
Expectation of a real random variable on a finite probability space
Definition
The expectation of a real random variable on a finite probability space is For a real constant , the notation also denotes the constant random variable .
Depends on
Used by
- Expectation agrees with the published finite weighted sum Corollary
- Expectation equal to 1 does not force a nonnegative integer-valued variable to vanish somewhere Counterexample
- For dependent variables, E[XY] need not equal E[X]E[Y] Counterexample
- Markov's conclusion can fail without nonnegativity Counterexample
- The moment generating function M_X(t)=E[e^tX] on a finite probability space Definition
- Variance, standard deviation, and covariance on a finite probability space 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
- A random inner linear code has fewer than one bad word in expectation Lemma
- Cauchy-Schwarz for finite random variables: E[XY]² leE[X²]E[Y²] Lemma
- Conditional expectation constructs the inner code deterministically Lemma
- Expectation is the sum of each attained value times its probability Lemma
- Indicators turn event probabilities, intersections, and finite counts into expectations and products Lemma
- Pairwise-independent hashing controls fibre size Lemma
- Sequential repetition amplifies completeness and soundness gaps Lemma
- Small total induced-copy expectation forces many homogeneous k-sets Lemma
- Variance and covariance identities for random variables Lemma
- Expectation is linear for every finite family of random variables, without any independence hypothesis Theorem
- Expectation preserves pointwise order and lies between the minimum and maximum attained values Theorem
- The finite second-moment bound ℙ(X≠0)geE[X]²/E[X²] when E[X²]>0 Theorem
Dependency tree · two levels
14 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 6.1 (standard reference, not scraped)
- M. Bucic, Probabilistic Method, Definition A.5 (standard reference, not scraped)