TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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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.

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Variance of a finite sum as the sum of all variances and covariances

Statement

For a finite family $(X_{i})_{i \in I}$ , $Var ⁣ (\sum_{i \in I} X_{i}) = \sum_{i \in I} Var (X_{i}) + 2 \sum_{{i, j} \subseteq I} Cov (X_{i}, X_{j}) .$ Equivalently, it is $\sum_{i, j \in I} Cov (X_{i}, X_{j})$ . The empty sum has variance zero, and the singleton formula is the identity. In the first display, the second sum is over two-element subsets of $I$ .

Facts & Assumptions

Given: A finite family of random variables $(X_{i})_{i \in I}$ .

[L1]

$Var (X) = Cov (X, X)$ (Variance, standard deviation, and covariance on a finite probability space).

[L2]

Covariance is symmetric and bilinear in finite linear combinations (Covariance is symmetric and bilinear in finite linear combinations).

Proof

technique · direct

1.1

By [L1] and bilinearity, $Var (\sum_{i} X_{i}) = Cov (\sum_{i} X_{i}, \sum_{j} X_{j}) = \sum_{i, j} Cov (X_{i}, X_{j})$ .

L1L2

2.1

Separate the diagonal terms, which are $Var (X_{i})$ , from the off-diagonal ordered pairs. Symmetry pairs the latter into twice the sum over unordered pairs.

step 1.1L1L2algebra

3.1

For an empty family every sum in step 1.1 is zero, and for a singleton only its diagonal term remains.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

7 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. Matousek and J. Vondrak, The Probabilistic Method, Section 6.1 (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 5.3.1 (standard reference, not scraped)