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.
Law or distribution of a random element
Definition
Let be a random element. Its law or distribution is the set function
Thus the law of records the probability of each measurable target set by pulling it back to an event in the original probability space.
Depends on
Used by
- Dominating thinning for bounded time-varying propensities Algorithm
- Gillespie direct method with horizon and budget statuses Algorithm
- Minimum-intensity coupling of reaction networks Algorithm
- Next-reaction method by internal Poisson clocks Algorithm
- Rejection sampling Algorithm
- Countably many independent copies of a prescribed law exist Corollary
- Posterior RCD exists for a standard-Borel parameter Corollary
- Cauchy averages admit no deterministic weak centering Counterexample
- Uniform payoff convergence does not imply gradient convergence Counterexample
- Assumptions for identifying natural effects Definition
- Bayesian posterior as a regular conditional law Definition
- Binding function and indirect-inference target Definition
- Chi-square laws, including zero degrees Definition
- Competing-risk data, cause-specific hazard measures, and cumulative incidence Definition
- Conditional law given a random element Definition
- Conditional mutual information by a same-marginal product law Definition
- Convergence in distribution for real random variables Definition
- Convergence in distribution of random elements Definition
- Cumulative distribution function of a real random variable Definition
- Exchangeable random elements Definition
- Finite randomized treatment-assignment models Definition
- Full and observed data under a missingness pattern Definition
- Ideal and finite-simulation synthetic likelihood Definition
- Identical distribution and IID families Definition
- Joint convergence in distribution and the normalized cycle-character observables Definition
- L2-m-approximable causal Bernoulli shifts Definition
- Measure-theoretic mutual information Definition
- Multivariate normal law, including singular covariance Definition
- Pivotal quantities Definition
- Process law, modification, and indistinguishability Definition
- Random fields, stationarity, covariance, and variograms Definition
- Random measures and locally finite point processes Definition
- Scientific statistical model contracts Definition
- Standard normal and normal laws Definition
- Statistical experiments Definition
- Strict and weak stationarity for integer-indexed time series Definition
- Student t and Fisher F laws Definition
- Symmetric real random variables Definition
- Wishart law by Gaussian outer products Definition
- A countable probability space with geometric weights Example
…and 39 more results.
Dependency tree · two levels
6 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
- Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.2 (standard reference, not scraped)
- J. R. Norris, Probability and Measure, Section 2.2 (standard reference, not scraped)