Variance, standard deviation, and covariance on a finite probability space

Depends on

• Expectation of a real random variable on a finite probability space
• The reals form a totally ordered field
• Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$

Used by

• Uncorrelated finite random variables need not be independent Counterexample
• Stratified interference and source design-based variance estimators Definition
• A finite-history conditional mean can be nonlinear Example
• A rank-deficient universal-kriging design Example
• A singular history covariance can have nonunique coefficients Example
• A symmetric two-point distribution attains equality in Chebyshev's inequality Example
• An atomic sinusoid and an AR(1) process have different spectra Example
• Ordinary kriging can have larger error than simple kriging Example
• Pairwise uncorrelated data can be jointly dependent Example
• Singular observation covariance in simple kriging Example
• SRSWOR inclusion probabilities, pairwise nonpositive indicator covariance, and the sample mean Example
• The binomial sample proportion Example
• The expected number of triangles in G(n,p) is C(n, 3)p³ Example
• Finite simple-random-sampling moment identities Lemma
• Finite two-stage sampling variance identities Lemma
• Pairwise-independent hashing controls fibre size Lemma
• Typical-projector bounds for finite quantum sources and cq ensembles Lemma
• Var(X)=E[X²]-E[X]² and Cov(X,Y)=E[XY]-E[X]E[Y] Lemma
• Variance and covariance identities for random variables Lemma
• Chebyshev's inequality on a finite probability space Theorem
• Conditional hypergeometric moments for a logrank risk set Theorem
• Stratified-interference variance estimators are conservative 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

• C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 6.2 (standard reference, not scraped)
• H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 5.3.1 (standard reference, not scraped)