Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 7 / a / iii

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- Solution iii

Solution

The conditional expectation given an event of positive probability is

E (X ∣ B_{i}) = j = 0 \sum \infty j P (X = j ∣ B_{i}) .

(1)

Apply the law of total probability to each event

{X = j}

E (X) = j = 0 \sum \infty j i = 1 \sum k P (X = j ∣ B_{i}) P (B_{i}) = i = 1 \sum k P (B_{i}) j = 0 \sum \infty j P (X = j ∣ B_{i}) = i = 1 \sum k P (B_{i}) E (X ∣ B_{i}) .

(2)

This is the law of total expectation. Interchanging the sums is justified by nonnegative terms, for example by Tonelli theorem, so it remains valid when the expectation is infinite.

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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 (resit) / 7 / a / iii

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