Lecture 10 Conditional Expectation

We say that P[A|B] the conditional probability of A, given B. It is important to note that the condition P[B] > 0 is crucial. When X and Y are random variables defined on the same probability space, we often want to give a meaning to the expression P[X ∈ A|Y = y], even though it is usually the case that P[Y = y] = 0. When the random vector (X, Y) admits a joint density fX,Y(x, y), and fY(y) > 0, the concept of conditional density fX|Y=y(x) = fX,Y(x, y)/ fY(y) is introduced and the quantity P[X ∈ A|Y = y] is given meaning via ∫ A fX|Y=y(x, y) dx. While this procedure works well in the restrictive case of absolutely continuous random vectors, we will see how it is encompassed by a general concept of a conditional expectation. Since probability is simply an expectation of an indicator, and expectations are linear, it will be easier to work with expectations and no generality will be lost. Two main conceptual leaps here are: 1) we condition with respect to a σ-algebra, and 2) we view the conditional expectation itself as a random variable. Before we illustrate the concept in discrete time, here is the definition.