This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Pstat160a Handout 2 Conditioning I. Conditional p.d.f Recall the definition of the conditional p.d.f For X, given Y = j : p X  Y = j ( i ) = p X,Y ( i,j ) p Y ( j ) j fixed, for Y, given X = i : p Y  X = i ( j ) = p X,Y ( i,j ) p X ( i ) i fixed Just as for usual conditional probability of events the “Bayes rules” states that for discrete random variables p X,Y ( i,j ) = p X  Y = j ( i ) p Y ( j ) p X ( i ) = X j p X  Y = j ( i ) p Y ( j ) (1) p Y ( j ) = X i p Y  X = i ( j ) p X ( i ) Example 1: In chapter 1 in class, example 3) (see also homework 2, problem 4) we looked at the joint pdf found that the conditional pdf p X,Y ( i,j ) = P ( X = i, Y = j ) = p (1 p ) i 1 i for i ≥ 1 , 1 ≤ j ≤ i. otherwise And we found that the marginal of X was Geometric ( p ) and that the conditional pdf of Y given X = i was uniform on 1 , 2 , ··· ,i : p Y  X = i ( j ) = p (1 p ) i 1 i p (1 p ) i 1 = 1 i for j = 1 , ··· ,i II. Conditional expectation Notice that the conditional pdf is a pdf itself (in the variable that is not conditioned upon). Once can therefore define the conditional expectation (or conditional expected value) as E ( X  Y = j ) = X i ip X  Y = j ( i ) E ( Y  X = i ) = X j jp Y  X = i ( j ) or more generally E ( g ( X )  Y = j ) = X i g ( i ) p X  Y = j ( i ) Going back to Example 1 , using the fact that the expected value of a discrete uniform on 1 , ··· ,n is n +1 2 , we get E ( Y  X = i ) = i + 1 2 Remark: Note that the conditional expectation E ( Y  X = i ) is a function of i , the value that X took. However, X is a random variable, so we could think of the conditional expectation as a function of the random variable X , and write E ( Y  X ). In Example 1 , we get E ( Y  X ) = X + 1 2 and note that E ( Y  X ) is itself a random variable. III. Tower property and first applications 1 2 As notice above, we can think of E ( Y  X ) as a random variable. What is its expected value? Since E ( Y  X = i ) is just a function of i , the variable of X , or, which is the same thing, E ( Y  X ) is a function of X (say E ( Y  X ) = g ( X ), the expected value needs to be taken with respect to the pdf of X (see chapter 1). The result is the socalled(see chapter 1)....
View
Full
Document
This note was uploaded on 01/10/2011 for the course STAT 160A taught by Professor Bonnet during the Winter '10 term at UCSB.
 Winter '10
 bonnet

Click to edit the document details