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

A partition of a set $\Omega$ is a collection of nonempty subsets $B_i$ such that $B_i\cap B_j=\varnothing$ for $i\ne j$ and ${\bigcup }_{i\in I}{B}_i=\Omega$. Thus every outcome lies in exactly one cell. For a probability-space partition the cells must also be measurable events. Nonempty cells may nevertheless have probability zero.

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 $+\infty$.