Conditional Expectation and Variance#

Introduction#

The conditional probability of a random variable \(X\) given a random variable \(Y\) is a legitimate probability in terms of \(X\), and consequently, we can use it to compute the conditional expectation of \(X\) given \(Y\). A simple example is that we have a population of people, consisting of male and females, alongside with their weights. It may be more useful to know the average weight of a particular gender than the average weight of the population. More concretely, let \(X\) be the weight of a person and \(Y\) be the gender, then we are interested in the \(\mathbb{E}[X|Y]\) rather than just \(\mathbb{E}[X]\).

Further Readings#