Stat 5101 Lecture Notes - School of Statistics

3.3. AXIOMS FOR CONDITIONAL EXPECTATION 89Axiom CE1. If Y is in L 1 and a is any function, thenE{a(X)Y | X} = a(X)E(Y | X).We don’t have to verify that conditional expectation obeys the axioms ofordinary unconditional expectation, because conditional expectation is a specialcase of unconditional expectation (when thought about the right way), but thisaxiom isn’t a property of unconditional expectation, so we do need to verifythat it holds for conditional expectation as we have already defined it. But theverification is easy.∫E{a(X)Y | X} = a(X)yf(y | X) dy∫= a(X) yf(y | X) dy= a(X)E(Y | X)because any term that is not a function of the variable of integration can bepulled outside the integral (or sum in the discrete case).Two comments:• We could replace big X by little x if we wantE{a(x)Y | x} = a(x)E(Y | x)though, of course, this now follows from Axiom E2 of ordinary expectationbecause a(x) is a constant when x is a constant.• We could replace big Y by any random variable, for example, g(Y ) forany function g, obtainingE{a(X)g(Y ) | X} = a(X)E{g(Y ) | X}.3.3.2 The Regression FunctionIt is now time to confront squarely an issue we have been tiptoeing aroundwith comments about writing E(Y | x) orE(Y |X) “according to taste.” Inorder to clearly see the contrast with unconditional expectation, let first reviewsomething about ordinary unconditional expectation.E(X) is not a function of X. It’s a constant, not a random variable.This doesn’t conflict with the fact that an expectation operator is a functionE : L 1 → R when considered abstractly. This is the usual distinction between afunction and it’s values: E is indeed a function (from L 1 to R), but E(X) isn’ta function, it’s the value that the expectation operator assigns to the randomvariable X, and that value is a real number, a constant, not a random variable(not a function on the sample space).So E(X) is very different from g(X), where g is an ordinary function. Thelatter is a random variable (any function of a random variable is a randomvariable).So what’s the corresponding fact about conditional expectation?