Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13
How 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 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
- Cauchy-Schwarz for finite random variables: E[XY]² leE[X²]E[Y²] 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
- 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 · next 3 levels
Direct dependencies and their dependencies through the next three levels: 34 results over 14 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
- 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)