lecture7 - ISYE 2028 A and B Lecture 7 Conditional...

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Unformatted text preview: ISYE 2028 A and B Lecture 7 Conditional Expectation and Prediction Dr. Kobi Abayomi February 10, 2009 1 Conditional Expectation (again) We should recall the definition of the conditional expectation: E ( X | Y = y ) = x x P ( x | Y = y ) , discrete R xf X | Y = y dx, continuous (1) In the case that (either) p y ,f y are functions of only y (i.e. when Y 6 = g ( X ): E ( X | Y = y ) = ( P x x P ( x,y ) p y , discrete R xf X | Y = y dx f Y , continuous (2) If you are in doubt i.e. you are trying to solve a problem calculate f Y = R f X,Y dx explicitly. 1.1 Example Let X,Y f X,Y = e- x/y e- y y 1 { x,y (0 , ) } . Find the conditional expectation E ( X | Y ). Well: 1 f Y = Z e- x/y e- y y dx = e- y Z 1 y e- x/y dx = e- y Which yields... f X | Y = e- x/y e- y y 1 e- y = 1 y e- x/y So, the conditional expectation is: E ( X | Y = y ) = Z x y e- x/y dx But this is just the expectation of an exponential random variable with parameter = 1 y . So E ( X | Y ) = y . 2 Computing Expectations by Conditioning Recall: E ( X ) = E ( E ( X | Y )); the expectation of the conditional expectation is the original, or unconditioned expectation. We saw a proof of this in an earlier lecture. This fact allows us to compute expectations by conditioning on any random variable (which makes the computation easy). 2.1 Example: Expectation of a Random Sum Let X 1 ,...,X N with N a random number and X i N . Find the expectation of the random sum E ( N i =1 X i ). Condition on N : 2 E ( N X i =1 X i ) = E [ E ( N X i =1 X i | N = n )] = E [ N E ( X )] = E ( N ) E ( X ) 2.2 Example: in N 2 is correlation Let X,Y N 2 ( X , Y , 2 X , 2 Y , ), the bivariate normal distribution. Weve defined the correlation X,Y...
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This note was uploaded on 11/08/2009 for the course ISYE 2028 taught by Professor Shim during the Spring '07 term at Georgia Institute of Technology.

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lecture7 - ISYE 2028 A and B Lecture 7 Conditional...

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