This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
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.
 Spring '07
 SHIM

Click to edit the document details