Lecture_11

Lecture_11 - 1 CONDITIONAL EXPECTATION Lecture 11...

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1 CONDITIONAL EXPECTATION Lecture 11 ORIE3500/5500 Summer2009 Chen Class Today Conditional Expectation Variance and Standard Deviation Covariance 1 Conditional Expectation We have deﬁned expectation of a random variable only in two cases: discrete and continuous. Expectation can be deﬁned for all random variables but that is outside the scope of this class. We will deﬁne conditional expectation in these two cases only. 1. If ( X,Y ) is a discrete bivariate random vector with joint pmf p X,Y ( x i ,y j ), then the conditional expectation of X given Y = y j is deﬁned to be E ( X | Y = y j ) = X i x i p X | Y ( x i | y j ) . 2. If ( X,Y ) is a continuous bivariate random vector with joint pdf f X,Y ( x,y ), then the conditional expectation of X given Y = y is deﬁned to be E ( X | Y = y ) = Z -∞ xf X | Y ( x | y ) . Let a function be g ( y ) = E ( X | Y = y ), observe that g ( Y ) is also a random variable. We could deﬁne its expectation E ( g ( Y )) = E [ E ( X | Y = y )] j E [ X | Y = y j ] p Y ( y j ) Y is discrete R -∞ E [ X | Y = y ] f Y ( y ) dy Y is continuous Law of Iterated Expectation: E ( X ) = E [ E ( X | Y )] 1

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2 VARIANCE AND STANDARD DEVIATION Example Here is a game: there are two diﬀerent coins. One is fair and the other one has 0 . 9 chance to get tails when it is tossed. If you have 3 / 4 probability to get the fair one and you will win one dollar if you get a head and nothing otherwise. What is the expectation of the dollar you could win? Deﬁne the random variable X as the dollar you could win, and deﬁne a random variable Y , which equals to one if you pick the fair coin and zero otherwise. Then we have E [ X | Y = 1] = 1 · 1 2 + 0 · 1 2 = 0 . 5 , and E [ X | Y = 0] = 1 · 0 . 1 + 0 · 0 . 9 = 0 . 1 . Finally,
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Lecture_11 - 1 CONDITIONAL EXPECTATION Lecture 11...

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