Stat 5101 Lecture Notes - School of Statistics

3.4. JOINT, CONDITIONAL, AND MARGINAL 953.4 Joint, Conditional, and MarginalAs was the case with unconditional expectation, our “axioms first” treatmentof conditional expectation has been a bit abstract. When the problemis solved by pulling a function of the conditioning variables outside of a conditionalexpectation or by the iterated expectation formula, either the specialcase in Axiom CE2 with the outside expectation an unconditional one or thegeneral case in Theorem 3.3 in which both expectations are conditional, thenthe axioms are just what you need. But for other problems you need to be ableto calculate conditional probability densities and expectations by doing sumsand integrals, and that is the subject to which we now turn.3.4.1 Joint Equals Conditional Times MarginalNote that the iterated expectation axiom (Axiom CE2), when we write outthe expectations as integrals, equates∫ (∫)E{E(Y | X)} = yf(y | x) dy f X (x) dx∫∫(3.12a)= yf(y | x)f X (x) dx dyand∫∫E(Y )=yf(x, y) dx dy.(3.12b)Equation (3.12b) is correct, because of the general definition of expectation ofa function of two variables:∫∫E{g(X, Y )} = g(x, y)f(x, y) dx dywhenever the expectation exists. Now just take g(x, y) =y.One way that the right hand sides of (3.12a) and (3.12b) can be equal is iff(x, y) =f(y|x)f X (x) (3.13)or in words,joint = conditional × marginalIn fact, by the uniqueness theorem (Theorem 3.2), this is the only way theiterated expectation axiom can hold, except, as usual, for possible redefinitionon sets of probability zero.This gives a formula for calculating a conditional probability density fromthe jointf(x, y)f(y | x) = (3.14)f X (x)or in words,conditional =jointmarginal