---
slug: expected_value
name: Expected Value
one_liner: "The probability-weighted average of a random quantity — the single scalar a distribution 'amounts to' in the long run, computed by the linear expectation operator E[·] as an integral/sum of value times probability, and the canonical bridge from a distribution to a decision-relevant number."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, mathematical, computational]
origin_domain: probability_theory
alternate_origin_domains: [decision_theory, statistics, finance, statistical_physics]
aliases: [expectation, mean, expectation_operator, probability_weighted_average, ensemble_average, long_run_average]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-06 modalization, flavor 6b deterministic→stochastic) — the GENUS obtained as the convergent alias-target of dozens of '6b' applications that each turned a deterministic prime into a distribution and then asked for the number that distribution amounts to (maneuver, antagonist, informal_enforcement, pivotality, switching_cost, technical_debt, ...). Every such input resolved to the same structure: the probability-weighted average via the expectation operator E[·]. Discovered 2026-07-03. MODERATE hold (verified as a large genus convergence; established mathematical name — the expectation operator)."
- "FOLD-VS-GENUS FLAG (PROMINENT — this is the dominant keep/cut question): `expected_utility` IS PRESENT in the corpus. expected_value is EITHER (a) the GENUS of which expected_utility is the special case (expected utility = expectation of a UTILITY function; expected value = expectation of ANY random quantity — money, counts, energy, time), OR (b) a fold into expected_utility. Proposer argues (a): the expectation operator is more general and more basic than expected utility, applies with no utility/decision frame, and many corpus primes invoke averaging-over-uncertainty with no utility involved. But the owner may prefer to fold, or to keep only the present utility special case. EVALUATE expected_value AND expected_utility TOGETHER."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "Phase-C action: re-similarity-check vs `expected_utility` FIRST (fold-vs-genus decision), then `probability`, `distributional_assumption`, `uncertainty`, `risk`; if kept as genus, draw a subsumption edge expected_value → expected_utility; write a v1 draft and sync the one_liner across v1/v2 per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 4
  domain_breadth: 5
  structural_abstraction: 5
  transfer_evidence: 5
  reasoning: 'The bare structure — a probability distribution, a value function on its outcomes, and a linear operator that collapses the distribution to a single representative scalar (an integral/sum of value times probability) — is a pure mathematical form carrying no domain substance. It recurs identically wherever a quantity is uncertain and one wants the number it "amounts to" on average: expected return in finance, ensemble/thermal average in statistical physics, expected count in queueing and reliability, expected loss in statistics and machine learning, and expected payoff in decision theory and game theory. These are structurally-unlike siblings sharing only the expectation operator, so domain_breadth scores 5 and transfer_evidence 5 (the operator is independently canonical in every one of these literatures). Structural abstraction is 5: the core is a content-free statement about a distribution and a value function, with a defining structural property — linearity, E[aX+bY]=aE[X]+bE[Y] — that a single deterministic value cannot possess. Composite sits at 4 only because the notion presupposes a well-defined probability measure and integrable value function, a mild formal precondition rather than a substantive frame.'
similarity_to_nearest_existing_prime:
  slug: expected_utility
  similarity: 0.66
structural_framed:
  label: structural
  aggregate: 0.30
  criteria: {vocab_travels: 0.8, evaluative_weight: 0.25, institutional_origin: 0.1, human_practice_bound: 0.15, import_vs_recognize: 0.35}
  rationale: 'Reads structural. The core — a distribution, a value function, and the linear operator that averages value against probability — is recognized rather than imported and travels cleanly into finance, physics, statistics, and decision theory without carrying any originating vocabulary. The frame residue is light: choosing a value function and speaking of a "decision-relevant" number import a mild evaluative cast (keeping evaluative_weight and import_vs_recognize off the floor), but the underlying object is a substrate-neutral mathematical operator. Lower framing than most framed-stratum candidates and lower than its decision-theoretic special case `expected_utility`, which carries an explicit preference/utility frame that expected_value does not.'
phase_b_passes_completed: []
---
# Expected Value

## Core Idea

Expected value is the probability-weighted average of a random quantity: the single number that a whole distribution "amounts to" in the long run. Its anatomy has three parts — a probability distribution over possible outcomes, a value function that assigns a number to each outcome, and the *expectation operator* E[·] that collapses the distribution to one representative scalar by integrating (or summing) value against probability. Where a deterministic quantity sits at one fixed value, a random quantity spreads across many; the expectation is the operator that answers "if this quantity is uncertain, what one number does it come to on average?" — with companion notions of variance and higher moments describing the *spread* around that number.

The feature present here but absent at a single deterministic value is the collapse itself, and its defining structural property: *linearity*. E[aX + bY] = aE[X] + bE[Y] holds regardless of how X and Y are related — a property a lone value cannot have, because a lone value has nothing to average and no distribution to be linear over. This is what makes the expectation the canonical *bridge from a distribution to a decision-relevant scalar*: modalizing a deterministic prime (op-06, flavor 6b) turns a fixed value into a distribution over values, and the expectation is the operator that maps that distribution back to a single actionable number. Dozens of such modalizations — the "expected maneuver," the probability-weighted "pivotality," the average "switching_cost" under uncertainty — each reduce to exactly this structure, which is why they converge on it as their common genus.

Read structurally, the same operator appears across genuinely unlike domains under different names. In *finance* it is expected return — the probability-weighted average payoff of an uncertain investment. In *statistical physics* it is the ensemble or thermal average — a macroscopic quantity computed as the expectation over a distribution of microstates. In *statistics and machine learning* it is expected loss or risk — the average error a model incurs over the data distribution, the thing empirical risk minimization approximates. In *decision theory and game theory* it is expected payoff. Across all of these, the shared abstract object is one and the same: a distribution, a value function on it, and the linear operator that averages the second against the first. Its decision-theoretic specialization — the expectation of a *utility* function specifically — is the present prime `expected_utility`; expected_value is the more general operator of which that is one case.

## Notes

**Provenance.** Produced by operator 06 (modalization), flavor 6b (deterministic→stochastic), not as a single lift but as a *genus convergence*: dozens of 6b applications each took a deterministic prime, replaced its fixed value with a distribution over values, and then asked for the number that distribution amounts to. Inputs as varied as `maneuver`, `antagonist`, `informal_enforcement`, `pivotality`, `switching_cost`, and `technical_debt` all resolved to the identical structure — the probability-weighted average via the linear expectation operator E[·] — and named it as their alias-target. Surfaced on 2026-07-03. Established mathematical construct: the expectation operator of probability theory (expected value / expectation / mean). Verified as a large convergence; manifest records a MODERATE hold, gated on the fold-vs-genus question below.

**For owner evaluation (keep/cut) — the fold-vs-genus question is the whole decision.** `expected_utility` **is already present in the corpus**, and it sits close (proposer similarity ~0.66). The sharp question is therefore *not* whether expected_value is real, but whether it is (a) the **genus** of which `expected_utility` is the special case, or (b) a **fold** into `expected_utility`. The proposer argues (a): the expectation operator is strictly *more general and more basic* than expected utility — it is the expectation of ANY random quantity (money, particle counts, energy, waiting times, error), whereas expected utility is the expectation of a *utility function* specifically, carrying an explicit preference/decision frame that the bare operator does not. Many corpus primes invoke averaging-over-uncertainty (ensemble averages, expected loss, expected return) with no utility frame at all; those are naturally children of expected_value, not of expected_utility. The honest case for (b): the similarity is high, and an owner who wants a lean corpus may judge that the decision-relevant special case (`expected_utility`) already carries the weight, treating "expected value of a non-utility quantity" as a routine application rather than a distinct prime. **Recommendation: evaluate expected_value and expected_utility as a pair.** If kept, the natural relationship is a subsumption edge `expected_value` → `expected_utility` (genus → decision-theoretic species).

**At incorporation.** Re-run similarity against `expected_utility` **first** (this is the fold-vs-genus decision and dominates all else), then against `probability`, `distributional_assumption`, `uncertainty`, and `risk` to confirm no further collapse. If kept as the genus: draw a subsumption edge `expected_value` → `expected_utility`, and add missing-edge notes flagging the averaging-over-uncertainty relationship to `probability`/`distributional_assumption` and the spread-companion relationship to any variance/risk prime. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
