 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/4/a/iii/solution

A partition of a sample space is a finite or countable family

(A_{j})

of pairwise disjoint events whose union is the whole sample space. If

X

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

This follows by summing

X 1_{A_{j}}

over the partition of a sample space; absolute integrability justifies the countable sum. For nonnegative

X

, the identity also holds with infinite expectations. Zero-probability parts contribute zero without requiring a conditional value on them.

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