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 / i

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- Solution i

Solution

For an integrable discrete random variable, the conditional expectation given an event

B

of positive probability is

E [X ∣ B] = x \sum x P (X = x ∣ B) = x \sum x \frac{P ({ X = x } \cap B )}{P ( B )} .

The sum ranges over the possible values of

X

; integrability ensures absolute convergence.

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