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Variance of a finite sum as the sum of all variances and covariances
Statement
For a finite family of square-integrable real random variables on one common probability space, Equivalently, it is . The empty sum has variance zero, and the singleton formula is the identity. The original finite-probability-space case is included, since every real random variable there is square-integrable and the covariance/variance definitions agree with the finite centered-product/square formulas. In the first display, the second sum is over two-element subsets of .
Facts & Assumptions
Given: A finite family of square-integrable real random variables on one common probability space.
Covariance is symmetric and bilinear in finite linear combinations (Covariance is symmetric and bilinear in finite linear combinations).
Proof
The finite sum is square-integrable by repeated use of . By [L1] and bilinearity, .
Separate the diagonal terms, which are , from the off-diagonal ordered pairs. Symmetry pairs the latter into twice the sum over unordered pairs.
For an empty family every sum in step 1.1 is zero, and for a singleton only its diagonal term remains.
Depends on
Used by
- Average cluster size does not determine the unequal-size design effect Counterexample
- The binomial sample proportion Example
- A Bernoulli(p) variable has mean p and variance p(1-p); a binomial(n,p) variable has mean np and variance np(1-p) Lemma
- KL prior mass gives a posterior denominator lower bound in probability Lemma
- Local KL and log-square mass bounds a non-IID posterior denominator Lemma
- Classical bounded-sample mean baseline Theorem
- Equal-size cluster design effect under exchangeable within-cluster covariance Theorem
- Horvitz–Thompson variance and pair-inclusion estimate Theorem
- Terminal normal limit for weighted logrank scores Theorem
- Unbiasedness and variance decomposition for the two-stage expansion total Theorem
- Unbiasedness and variance of the SRSWOR cluster-total estimator Theorem
- Unbiasedness and variance of the stratified total estimator Theorem
Dependency tree · two levels
5 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. Matoušek and J. Vondrák, The Probabilistic Method, CMU 15-850 Fall 2020 course copy, Lemma 5.1.3, PDF pp. 24–25 (standard reference, not scraped)
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 5.3.1 (standard reference, not scraped)