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.
Variance, standard deviation, and covariance on a finite probability space
Definition
For real random variables and on one finite probability space, define The standard deviation of is , using the unique nonnegative square root supplied by Square roots exist: a unique with ; the positives are . Variance is nonnegative because its defining random variable is pointwise nonnegative and expectation is a sum with nonnegative weights.
Depends on
Used by
- Uncorrelated finite random variables need not be independent Counterexample
- A symmetric two-point distribution attains equality in Chebyshev's inequality Example
- The expected number of triangles in G(n,p) is C(n, 3)p³ Example
- Pairwise-independent hashing controls fibre size 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
- Variance of a finite sum as the sum of all variances and covariances Theorem
Dependency tree · two levels
17 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)