The uniform probability space on a nonempty finite set

Depends on

• Finite probability spaces, outcome weights, events, and event probabilities
• The cardinality $\lvert A\rvert$ of a finite set
• The reals form a totally ordered field

Used by

• Identity-corrected random-permutation Monte Carlo p-value Algorithm
• Events with the same probability need not be independent Counterexample
• 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 uncorrected Monte Carlo permutation fraction can be zero and invalid Counterexample
• Uncorrelated finite random variables need not be independent Counterexample
• Unions of overlapping independent events need not remain independent Counterexample
• PCP verifier resources and deterministic proof strings Definition
• Randomized complete-block design Definition
• Randomized Latin-square design Definition
• Two-group logrank and weighted logrank scores Definition
• A full-period LCG can alternate its low bit Example
• A symmetric two-point distribution attains equality in Chebyshev's inequality Example
• Independent events need not be disjoint Example
• The union bound can be strict for overlapping events Example
• Three pairwise-independent events that are not mutually independent Example
• False: linearity of expectation requires independence False statement
• A successive-revelation variance bound for a uniform permutation Lemma
• Mixture coupling for permuted two-sample U-statistics Lemma
• Small total induced-copy expectation forces many homogeneous k-sets Lemma
• The RSK union bound localizes Plancherel profiles Lemma
• Tied upper ranks on a finite multiset Lemma
• Conditional hypergeometric moments for a logrank risk set Theorem
• Every nonempty finite set of n nonzero integers has a sum-free subset of size greater than n/3 Theorem
• Exact finite-group permutation p-values and tests Theorem
• The power-of-two mixed LCG has full period but dependent adjacent states Theorem
• The RSK shape of a uniform random permutation has the Plancherel law Theorem
• Validity of identity-corrected random-permutation Monte Carlo p-values 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

• J. Matousek and J. Vondrak, The Probabilistic Method, Section 1.1 (standard reference, not scraped)
• M. Bucic, Probabilistic Method, Example A.2 (standard reference, not scraped)