160a_ho2 - Pstat160a Handout 2 Conditioning I Conditional...

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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. 0 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
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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 so-called tower property : E E ( Y | X ) = E ( Y ) or E ( Y ) = X i =0 E ( Y | X = i ) p X ( i ) in developed form
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