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

For a nonnegative integer-valued random variable, its expectation is the probability-weighted sumAll terms are nonnegative, so this is always defined as an extended nonnegative number, possibly $+\infty$.

Because the $B_i$ partition the sample space, the events $A\cap B_i$ are disjoint and their union is $A$. Finite additivity and the definition of conditional probability give the law of total probabilityThe assumption $\mathbb{P}(B_i)>0$ ensures that each conditional probability is defined.

The conditional expectation given an event of positive probability isApply the law of total probability to each event $\{X=j\}$: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.