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Expectation agrees with the published finite weighted sum
Statement
Let be a finite probability space and let . After identifying with the full-power-set probability space of Finite probability spaces are exactly finite full-power-set probability spaces, the expectation defined by Expectation of a nonnegative or integrable random variable agrees with the published finite formulas:
Facts & Assumptions
Given: A finite probability space and a real-valued function .
Finite probability spaces are exactly full-power-set probability spaces, and every finite real random variable is measurable there (Finite probability spaces are exactly finite full-power-set probability spaces, Finite random variables are measurable).
Change of variables for expectation identifies with the integral of the identity function against the law of (Change of variables for expectation).
The published finite expectation is , and it is also the sum over attained values weighted by their probabilities (Expectation of a real random variable on a finite probability space, Expectation is the sum of each attained value times its probability).
Proof
By [L1], the general expectation and the law of are defined on the same full-power-set probability space attached to .
Applying [L2] to the identity map on gives For a finite random variable, [L3] identifies this quantity with both
Thus the general expectation is exactly the published finite weighted-sum expectation and its finite-distribution reformulation.
Depends on
Used by
- Unequal error variances can make OLS fail to be BLUE Counterexample
- Finite-alphabet conditional entropy given an observation Definition
- A data-dependent rule dominates a constant rule Example
- Bernoulli score, information, and attainment of the scalar bound Example
- Perfect correlation can coexist with a method offset Example
- Randomization lowers worst-case risk under zero-one loss Example
- Repeated method differences can share a random intercept Example
- Squared-error risk for a Bernoulli observation Example
- The mean has an unbounded directional derivative supremum Example
- Conditional expectation on positive-probability events in finite spaces Lemma
- Dyadic positive-mean estimator under an L2 bound Lemma
- Finite KL divergence agrees with measurable-space KL Lemma
- One-sided concentration bound for independent median trials Lemma
- Variance-aware quantum Monte Carlo guarantee Theorem
Dependency tree · two levels
23 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 1.6.3 (standard reference, not scraped)