Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 8 / a / ii

For an integrable random variable $X$, the law of total expectation on the finite or countable measurable partition isHere $\mathbb{E}(X\mid B_i)=\mathbb{E}[X{1}_{B_i}]/\mathbb{P}(B_i)$ for positive-probability cells; null cells contribute zero and need no conditional value. The series is absolutely convergent since its absolute terms sum to at most $\mathbb{E}|X|$. For a nonnegative $X$, the same identity is valid with possible value $+\infty$.