Stat 5101 Lecture Notes - School of Statistics

94 Stat 5101 (Geyer) Course Notes• The iterated expectation on the left hand side of (3.10) is the unique (upto redefinition on sets of probability zero) function f 2 (X) such thatE{g 2 (X)f 2 (X)} = E{g 2 (X)f 1 (X, Y )}(3.11b)for all functions g 2 such that g 2 (X)f 1 (X, Y ) ∈ L 1 .• E(Z | X) is the unique (up to redefinition on sets of probability zero)function f 3 (X) such thatE{g 3 (X)f 3 (X)} = E{g 3 (X)Z}(3.11c)for all functions g 3 such that g 3 (X)Z ∈ L 1 .Since (3.11a) holds for any function g 1 , it holds when g 1 (X, Y )=g 3 (X),from which, combining (3.11a) and (3.11c), we getE{g 3 (X)f 3 (X)} = E{g 3 (X)Z} = E{g 3 (X)f 1 (X, Y )}(3.11d)Reading (3.11d) from end to end, we see it is the same as (3.11b), because (3.11d)must hold for any function g 3 and (3.11b) must hold for any function g 2 .Thusby the uniqueness assertion of Theorem 3.2 we must have f 2 (X) =f 3 (X), exceptperhaps on a set of probability zero (which does not matter). Since f 2 (X) isthe left hand side of (3.10) and f 3 (X) is the right hand side, that is what wasto be proved.Theorem 3.2 can also be used to prove a very important fact about independenceand conditioning.Theorem 3.4. If X and Y are independent random variables and h is a functionsuch that h(Y ) ∈ L 1 , thenE{h(Y ) | X} = E{h(Y )}.In short, conditioning on an independent variable or variables is the sameas conditioning on no variables, making conditional expectation the same asunconditional expectation.Proof. If X and Y are independent, the right hand side of (3.9) becomesE{g(X)}E{h(Y )} by Definition 2.7.2. Hence, in this special case, Theorem 3.2asserts that E{h(Y ) | X} is the unique function f(X) such thatE{g(X)f(X)} = E{g(X)}E{h(Y )}whenever g(X) ∈ L 1 . Certainly the constant f(X) =a, where a = E{h(Y )} isone such function, becauseE{g(X)a} = E{g(X)}a = E{g(X)}E{h(Y )}so by the uniqueness part of Theorem 3.2 this is the conditional expectation, aswas to be proved.