Stat 5101 Lecture Notes - School of Statistics

88 Stat 5101 (Geyer) Course NotesRoman letter behind the bar; there we had a Greek letter behind the bar, but(mathematics is invariant under changes of notation) that makes no conceptualdifference whatsoever.The fact that conditional probability is a special case of ordinary probability(when we consider the variable or variables behind the bar fixed) means that wealready know a lot about conditional probability. Every fact we have learnedso far in the course about ordinary probability and expectation applies to itsspecial case conditional probability and expectation. Caution: What we justsaid applies only when the variable(s) behind the bar are considered fixed. Aswe shall see, things become more complicated when both are treated as randomvariables.Vector VariablesOf course, either of the variables involved in a conditional probability distributioncan be vectors. Then we write either off(y | x)f(y 1 ,...y n |x 1 ,...x m )according to taste, and similarly either ofE(Y | x)E(Y 1 ,...Y n |x 1 ,...x m )Since we’ve already made this point in the context of parametric families ofdistributions, and conditional probability distributions are no different, we willleave it at that.3.3 Axioms for Conditional ExpectationThe conditional expectation E(Y | x) is just another expectation operator,obeying all the axioms for expectation. This follows from the view explainedin the preceeding section that conditional expectation is a special case of ordinaryunconditional expectation (at least when we are considering the variableor variables behind the bar fixed). If we just replace unconditional expectationswith conditional expectations everywhere in the axioms for unconditionalexpectation, they are still true.There are, however, a couple of additional axioms for conditional expectation.Axiom E2 can be strengthened (as described in the next section), and anentirely new axiom (described in the two sections following the next) can beadded to the set of axioms.3.3.1 Functions of Conditioning VariablesAny function of the variable or variables behind the bar (the conditioningvariables) behaves like a constant in conditional expectations.