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.
Markov's inequality on a finite probability space
Statement
If is a nonnegative real random variable on a finite probability space and , then The weak threshold is part of the statement.
Facts & Assumptions
Given: A nonnegative real random variable and a real .
Expectation preserves pointwise order (Expectation preserves pointwise order and lies between the minimum and maximum attained values).
Proof
Pointwise, : on the event this is the threshold inequality, and off it the right side is zero while .
Taking expectations gives .
Division by gives the claimed inequality. The hypothesis is exactly what licenses the division.
Depends on
Used by
- Markov's conclusion can fail without nonnegativity Counterexample
- A parameter ledger for the high-girth, high-chromatic alteration proof Example
- A two-valued random variable attains equality in Markov's inequality Example
- First- and second-moment bounds for a nonempty Bernoulli random subset Example
- Chebyshev's inequality on a finite probability space Theorem
- For all positive k,ℓ, some finite graph has girth greater than ℓ and chromatic number greater than k Theorem
- For n≥1 independent random signs, ℙ(|S|≥ t)≤2 exp(-t²/(2n)) for t>0 Theorem
- The first-moment method for avoiding or forcing a finite count of bad events Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 29 results over 9 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
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 6.2.2 (standard reference, not scraped)
- J. Matousek and J. Vondrak, The Probabilistic Method, Theorem 4.1.1 (standard reference, not scraped)