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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability-resit/8/a/iii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 8 / a / iii / Solution by

Codex  0 2026-10-25

The finite-partition form of the law of total expectation states

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

(1)

Because the events form a partition of a sample space of the sample space,

\sum_{j} 1_{B_{j}} = 1

pointwise. Multiply by

X

and use linearity of expectation to obtain

E [X] = \sum_{j} E [X 1_{B_{j}}] = \sum_{j : P (B_{j}) > 0} P (B_{j}) E [X ∣ B_{j}] .

(2)

An event of zero probability contributes zero to the middle expression and is omitted from the final expression, so no undefined conditional expectation is multiplied by zero.

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