Conditional expectation

In this chapter, the authors introduce the general concept of a conditional expectation given a σ-algebra 𝒸. The concepts of a conditional expectation value and a conditional probability are defined more generally, using the factorization of a conditional expectation. The authors also define the general concept of a regression as a factorization g of a conditional expectation E(Y | X) = g(X), provided that X is real-valued. Furthermore, they define an (X=x)-conditional expectation value E(Y | X=x) as a value g(x) of the factorization g. This means that E(Y | X=x) is defined even if P(X=x) = 0. However, E(Y | X=x) is not uniquely defined. Nevertheless, they can formulate propositions about the conditional expectation values E(Y | X=x) for PX-almost all values x of X. Finally, they introduce the concept of mean independence and study its relationship to stochastic independence and uncorrelatedness.