LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-13

How 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.

Expectation is the sum of each attained value times its probability

Statement

If $X$ is a real random variable on a finite probability space, then $E [X] = \sum_{x \in X (Ω)} x P (X = x) .$

Facts & Assumptions

Given: A real random variable $X$ on $(Ω, w)$ .

[L1]

Expectation is $\sum_{ω \in Ω} X (ω) w (ω)$ (Expectation of a real random variable on a finite probability space).

[L2]

A finite sum may be split over disjoint fibres and reindexed (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).

Proof

technique · direct

1.1

The nonempty fibres $X^{- 1} ({x})$ , for $x \in X (Ω)$ , form a finite partition of $Ω$ .

given

2.1

Splitting the expectation over these fibres gives $E [X] = \sum_{x \in X (Ω)} \sum_{ω : X (ω) = x} x w (ω)$ .

step 1.1L1L2

3.1

The inner sum is $x P (X = x)$ , which proves the formula. The range cannot be empty because total probability is $1$ .

step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

9 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 6.1 (standard reference, not scraped)
M. Bucic, Probabilistic Method, Appendix A (standard reference, not scraped)