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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