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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Expectation preserves pointwise order and lies between the minimum and maximum attained values
Statement
If for every outcome, then . Consequently, for a real random variable , In particular some outcome has value at least , and some outcome has value at most .
Facts & Assumptions
Given: Real random variables on a finite probability space with pointwise.
Expectation is the finite weighted sum of a random variable, with nonnegative weights summing to (Expectation of a real random variable on a finite probability space).
Finite sums preserve pointwise order (Laws of finite sums and finite products).
Every nonempty finite list of reals has a maximum, and hence also a minimum by applying the maximum result to negatives (Every nonempty finite set of reals has a maximum and a minimum).
The real numbers form a totally ordered field (The reals form a totally ordered field).
Proof
Nonnegative weights give for every , so summing gives .
The outcome set is nonempty because its nonnegative weights sum to ; let and be the minimum and maximum of the attained values of .
The pointwise inequalities and step 1.1 give , since the expectations of the constant variables are and .
Outcomes attaining and therefore supply values at most and at least the expectation, including the constant case.
Depends on
Used by
- Cauchy-Schwarz for finite random variables: E[XY]² leE[X²]E[Y²] Lemma
- The deletion-alteration method converts an expected defect count into a deterministic lower bound Proposition
- An n-vertex graph of minimum degree δ>1 has a dominating set of size at most n(log(δ+1)+1)/(δ+1) Theorem
- Every finite graph with m edges has a cut containing at least m/2 edges Theorem
- Every nonempty finite set of n nonzero integers has a sum-free subset of size greater than n/3 Theorem
- Markov's inequality on a finite probability space Theorem
- Szele's bound: for every n≥1, some n-vertex tournament has at least n!/2ⁿ⁻¹ Hamilton paths Theorem
- The first-moment method for avoiding or forcing a finite count of bad events Theorem
Dependency tree · two levels
24 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, Appendix A (standard reference, not scraped)