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Conditional expectation on positive-probability events in finite spaces

Statement

Let $(Ω, w)$ be a finite probability space and $X : Ω \to R$ . For any event $B \subseteq Ω$ with $P (B) > 0$ , define $E [X ∣ B] : = \frac{1}{P (B)} \sum_{ω \in B} X (ω) w (ω) .$ If $A \subseteq Ω$ has $0 < P (A) < 1$ , then the function equal to $E [X ∣ A]$ on $A$ and $E [X ∣ A^{c}]$ on $A^{c}$ is a version of $E [X ∣ σ (A)]$ under the integral characterization. More generally, if $A_{1}, \dots, A_{r}$ is a finite partition of $Ω$ into events of positive probability, then $E [X] = \sum_{q = 1}^{r} P (A_{q}) E [X ∣ A_{q}] .$

Facts & Assumptions

Given: A finite probability space $(Ω, w)$ , a real function $X$ on $Ω$ , an event $A$ with $0 < P (A) < 1$ , and a finite positive-probability partition $A_{1}, \dots, A_{r}$ of $Ω$ .

[F1]

Finite probability spaces, outcome weights, events, and event probabilities: $Ω$ is finite, weights are nonnegative and sum to one, and event probabilities are sums of weights.

[F2]

Expectation of a real random variable on a finite probability space: $E [X] = \sum_{ω \in Ω} X (ω) w (ω)$ .

[F3]

Conditional expectation given a sigma algebra: a version given $σ (A)$ is measurable with respect to that sigma-algebra and has the same integral as $X$ on every event in it.

[F4]

Expectation agrees with the published finite weighted sum: on a finite probability space, the integral of a finite-valued function equals its finite weighted sum; applying this to $1_{E} X$ gives the integral of $X$ over any event $E$ .

[F5]

Laws of finite sums and finite products: finite sums distribute over addition and real scaling.

[F6]

Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule: sums over finite sets may be split over disjoint unions, and their values are invariant under bijective reindexing.

Proof

1.1F1F2F5givenalgebra

For any event $B$ with $P (B) > 0$ , the displayed definition is a finite real sum divided by a positive real. Its conditional weights $w_{B} (ω) = w (ω) / P (B)$ on $B$ are nonnegative and sum to one, since $\sum_{ω \in B} w_{B} (ω) = P (B) / P (B) = 1$ . Thus this formula is the expectation under the probability law restricted and normalized to $B$ .

2.1F1F2F3F4F6step 1.1given

Define $T (ω) = E [X ∣ A]$ for $ω \in A$ and $T (ω) = E [X ∣ A^{c}]$ for $ω \in A^{c}$ . The four events in $σ (A)$ are $\emptyset, A, A^{c}, Ω$ , so $T$ is $σ (A)$ -measurable. For either cell $B \in {A, A^{c}}$ , its finite weighted integral satisfies $\int_{B} T d P = P (B) E [X ∣ B] = \sum_{ω \in B} X (ω) w (ω) = \int_{B} X d P$ by step 1.1 and [F4]. Splitting over the cells using [F6] proves the same identity for every event in $σ (A)$ . Since $Ω$ is finite, $X$ and $T$ have finite weighted absolute sums and are integrable. Therefore [F3] makes $T$ a conditional-expectation version.

2.2F2F5F6step 1.1givenalgebra

For the partition $A_{1}, \dots, A_{r}$ , disjoint splitting [F6] gives $E [X] = \sum_{q = 1}^{r} \sum_{ω \in A_{q}} X (ω) w (ω)$ . By step 1.1, each inner sum is $P (A_{q}) E [X ∣ A_{q}]$ , proving the partition identity.

3.1step 1.1step 2.1step 2.2given∎

All conditional values and sums above are finite and every denominator is explicitly positive, so the proof uses no general conditional-expectation existence theorem or choice axiom.

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Sources

Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, 2nd ed., Chapter 4 §4.1 (standard reference, not scraped)
Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, 2nd ed., Chapter 6 §6.1, Definition 6.2 and Theorem 6.5 (standard reference, not scraped)