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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability/8/a/ii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 8 / a / ii / Solution by

Codex  0 2026-10-31

For an integrable random variable

X

, the law of total expectation on the finite or countable measurable partition is

E (X) = i : P (B_{i}) > 0 \sum P (B_{i}) E (X ∣ B_{i}) .

(1)

Here

E (X ∣ B_{i}) = E [X 1_{B_{i}}] / P (B_{i})

for positive-probability cells; null cells contribute zero and need no conditional value. The series is absolutely convergent since its absolute terms sum to at most

E ∣ X ∣

. For a nonnegative

X

, the same identity is valid with possible value

+ \infty

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