CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The expectation of an indicator is the probability of the event

Statement

Let $(Ω, F, P)$ be a probability space and let $A \in F$ . Then the indicator $1_{A}$ satisfies $E [1_{A}] = P (A) .$

Facts & Assumptions

Given: A probability space $(Ω, F, P)$ and an event $A$ .

[L1]

Expectation of a nonnegative random variable is its integral with respect to $P$ (Expectation of a nonnegative or integrable random variable).

[L2]

The complement identity gives $P (A^{c}) = 1 - P (A)$ (Basic identities for a probability measure).

[L3]

On nonnegative simple functions, the nonnegative integral is the simple integral $\sum_{j} c_{j} μ (E_{j})$ (The nonnegative integral agrees with the simple integral on simple functions, The integral of a nonnegative simple function).

Proof

technique · direct

1.1

The function $1_{A}$ is the simple function $1_{A} = 1 \cdot χ_{A} + 0 \cdot χ_{A^{c}} .$ Therefore [L1] and [L3] give $E [1_{A}] = 1 \cdot P (A) + 0 \cdot P (A^{c}) .$

L1L3

2.1

Using [L2] only to note that $A^{c}$ is the complementary event in the same probability space, step 1.1 simplifies to $E [1_{A}] = P (A)$ .

step 1.1L2∎

Depends on

Used by

The second-moment lower bound for positive probability Corollary

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

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.6 (standard reference, not scraped)