Stat 5101 Lecture Notes - School of Statistics

3.3. AXIOMS FOR CONDITIONAL EXPECTATION 93one: taking g = a and h the identity function in (3.8) gives our Axiom CE1,and taking g = 1 and h the identity function in (3.8) gives our Axiom CE2.Thus our treatment characterizes the same notion of conditional probability asstandard treatments.Another aspect of advanced treatments of conditional probability is thatstandard treatments usually take the statement Theorem 3.1 as a definitionrather than an axiom. The subtle difference is the following uniqueness assertion.Theorem 3.2. If X and Y are random variables and h is a function such thath(Y ) ∈ L 1 , then then there exists a function f such that f(X) ∈ L 1 andE{g(X)f(X)} = E{g(X)h(Y )} (3.9)for every function g such that g(X)h(Y ) ∈ L 1 . The function f is unique up toredefinition on sets of probability zero.The proof of this theorem is far beyond the scope of this course. Havingproved this theorem, advanced treatments take it as a definition of conditionalexpectation. The unique function f whose existence is guaranteed by the theoremis defined to be the conditional expectation, that is,E{h(Y ) | X} = f(X).The theorem makes it clear that (as everywhere else in probability theory)redefinition on a set (event) of probability zero makes no difference.Although we cannot prove Theorem 3.2, we can use it to prove a fancyversion of the iterated expectation formula.Theorem 3.3. If Y ∈ L 1 , thenE { E(Z | X, Y ) ∣ ∣ X } = E(Z | X). (3.10)Of course, the theorem also holds when the conditioning variables are vectors,that is, if m