Finite probability spaces, outcome weights, events, and event probabilities

Depends on

• A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$; $f : A \to B$, the value $f(a)$, domain and codomain
• The cardinality $\lvert A\rvert$ of a finite set
• Finite sums and finite products, by recursion
• The reals form a totally ordered field

Used by

• Finite rejection sampler for a linear combination Algorithm
• Size-n bootstrap from the empirical law Algorithm
• The uncorrected Monte Carlo permutation fraction can be zero and invalid Counterexample
• Without null invariance a group-rank p-value can be anti-conservative Counterexample
• Conditional probability ℙ(A∣ B) for ℙ(B)>0 Definition
• Expected query cost from finite execution prefixes Definition
• Finite conditional entropy Definition
• Finite conditional mutual information Definition
• Finite distributions, entropy, divergence, and stochastic matrices Definition
• Finite expectation-estimation task Definition
• Finite mutual information Definition
• Finite populations and sampling designs Definition
• Finite POVM outcome kernel Definition
• Finite randomized treatment-assignment models Definition
• Finite type-II testing function with explicit orientation Definition
• Finite-group randomization setups Definition
• Finite-population potential outcomes under partial interference Definition
• IID empirical bootstrap sample and mean statistic Definition
• Independent events, pairwise independence, and mutual independence of a finite family Definition
• Independent Local Quantum Fault Model Definition
• ℓ2 sample/query access to vectors and matrices Definition
• Local-Stochastic Fault-Set Law Definition
• Random Simon-oracle distribution for a classical lower bound Definition
• Randomized complete-block design Definition
• Randomized Latin-square design Definition
• Real random variables on finite probability spaces and their finite distributions Definition
• Simon independent-sample rank process Definition
• The finite product of finite probability spaces Definition
• The uniform probability space on a nonempty finite set Definition
• Two-stage randomized design and strategy-specific effects Definition
• A binary Neyman–Pearson boundary randomization Example
• A failed stationary covariance and a valid quadratic variogram Example
• A finite-history conditional mean can be nonlinear Example
• A loaded die as a nonuniform finite probability space Example
• A permutation orbit with tied ranks and nontrivial stabilizers Example
• A rank-deficient universal-kriging design Example
• A singular history covariance can have nonunique coefficients Example
• A tied two-sample Bernoulli permutation test Example
• A two-valued random variable attains equality in Markov's inequality Example
• An atomic sinusoid and an AR(1) process have different spectra Example

…and 59 more results.

Dependency tree · two levels

32 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 1.2 (standard reference, not scraped)
• H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Sections 1.3.2-1.3.3 (standard reference, not scraped)