Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 7 / a

A partition of a sample space is a finite or countable family of nonempty events such thatThus exactly one event in the family occurs in each outcome. To use ordinary event-conditioned conditional expectations, one usually assumes $\mathbb{P}(B_i)>0$; any zero-probability blocks can instead be omitted from the expectation formula.

For an integrable random variable $X$ and a partition of a sample space $(B_i)$, the law of total expectation statesIt follows by writing $X={\sum }_{i}X{1}_{B_i}$ and using $\mathbb{E}[X{1}_{B_i}]=\mathbb{P}(B_i)\mathbb{E}[X\mid B_i]$. Integrability justifies the countable sum. The same equality is valid with nonnegative, possibly infinite expectations when $X\ge 0$.