Stat 5101 Lecture Notes - School of Statistics

3.4. JOINT, CONDITIONAL, AND MARGINAL 103Recall from p. 90 in these notesSanity Check: E(X | Y ) is a function of Y and is not a functionof X.Good. We did get a function of y. If you get confused about which variableto integrate with respect to, this sanity check will straighten you out. If youthrough some mistake get a function of both variables, this sanity check will atleast tell you that you messed up somewhere.3.4.5 Bayes RuleNow we want to study the consequences ofjoint = conditional × marginal (3.26)Again we have the problem of remembering which marginal. If we recall ouranalysis ofconditional =jointmarginalon p. 100 in these notes, we recall that it is the marginal of the variable “behindthe bar.”Because “mathematics is invariant under changes of notation” (3.26) is alsotrue when we interchange the roles of the variables Hence we can “factor” ajoint density into marginal and conditional two different waysPlugging (3.27) into (3.21) givesf(x, y) =f(x|y)f Y (y) (3.27)f(x, y) =f(y|x)f X (x) (3.28)f(y | x) =f(x|y)f Y(y)∫f(x|y)fY (y)dy(3.29)This equation is called Bayes rule. It allows us to “turn around” conditionalprobabilities. That is, it is useful for problems that say: given f(x | y), findf(y | x). Or vice versa. Of course, because “mathematics is invariant underchanges of notation” (3.29) is also true with all the x’s and y’s interchanged.Example 3.4.6.Suppose that X and Y are positive real-valued random variables andwhat is f(y | x)?f(x | y) = 1 2 x2 y 3 e −xyf Y (y) =e −y