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Expectation of a countably valued nonnegative random variable is its weighted series

Statement

Let $(Ω, F, P)$ be a probability space. Let $(E_{j})_{j \geq 1}$ be a countable measurable partition of $Ω$ , let $c_{j} \in [0, \infty)$ be finite real numbers, and let $Z : Ω \to [0, \infty)$ be measurable with $Z = c_{j}$ on $E_{j}$ . Then, as an extended nonnegative value,

E [Z] = \sum_{j = 1}^{\infty} c_{j} P (E_{j}) .

If this series is finite, then $Z$ is integrable and its expectation is that finite sum.

Statement references

Probability measures and probability spaces: a probability measure has total mass one.

Expectation of a nonnegative or integrable random variable: expectation of a nonnegative measurable random variable is its Lebesgue integral.

Integrable real and complex functions, and their integrals: a real measurable function is integrable exactly when the integral of its absolute value is finite.

Linearity, monotonicity, and the modulus bound for expectation: expectation is linear on integrable real random variables.

The expectation of an indicator is the probability of the event: the expectation of an event indicator equals its probability.

Monotone convergence for the integral: integrals of increasing nonnegative measurable functions converge to the integral of their pointwise limit.

Series in the nonnegative extended real line: a nonnegative extended series is the supremum of its finite partial sums, including when the value is $+ \infty$ .

Facts & Assumptions

Given: A probability space, a countable measurable partition $(E_{j})_{j \geq 1}$ , finite constants $c_{j} \geq 0$ , and a measurable $Z$ equal to $c_{j}$ on each $E_{j}$ .

[F1]

If measurable nonnegative functions increase pointwise to $Z$ , their integrals increase to $\int Z d P$ (Monotone convergence for the integral).

[F2]

On a probability space, expectation of a nonnegative measurable variable is its nonnegative Lebesgue integral; a real measurable variable is integrable when the integral of its absolute value is finite; a nonnegative extended series is the supremum of its partial sums (Probability measures and probability spaces, Expectation of a nonnegative or integrable random variable, Integrable real and complex functions, and their integrals, Series in the nonnegative extended real line).

[F3]

Event indicators have expectation equal to event probability, and expectation is linear on finite sums of integrable real variables (The expectation of an indicator is the probability of the event, Linearity, monotonicity, and the modulus bound for expectation).

Proof

1.1givenF2construct

For $N \geq 1$ , define $Z_{N} : = \sum_{j = 1}^{N} c_{j} 1_{E_{j}}$ . This is a measurable nonnegative simple function. On $E_{j}$ , $Z_{N} = 0$ for $N < j$ and $Z_{N} = c_{j} = Z$ for $N \geq j$ , so $Z_{N} ↑ Z$ . Each $Z_{N}$ is bounded by the finite maximum of $c_{1}, \dots, c_{N}$ , hence is integrable on the probability space.

2.1step 1.1F3algebra

By linearity and the indicator identity in [F3], $E [Z_{N}] = \sum_{j = 1}^{N} c_{j} P (E_{j})$ .

3.1step 1.1step 2.1F1F2

Apply monotone convergence [F1] to step 1.1 and then use step 2.1. The resulting supremum of finite partial sums is exactly the nonnegative extended sum $\sum_{j = 1}^{\infty} c_{j} P (E_{j})$ by [F2], proving the extended-valued identity.

4.1givenF2step 3.1∎

If the series is finite, step 3.1 gives $\int Z d P < \infty$ . Since $Z \geq 0$ , $∣ Z ∣ = Z$ , so [F2] shows $Z$ is integrable and its ordinary expectation is the same finite sum.

Depends on

Used by

A finite-mean polynomial-tail reward is not sub-Gaussian Example

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Sources

John K. Hunter, Measure Theory Notes (standard reference, not scraped)