# tut07 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2005) Tutorial 7 Week of March 28, 2005 1. This problem requires significant thought but essentially no calculations. Let X and Y be two random variables. Let g ( y ) = E [ X | Y = y ] a = E [( g ( Y ) X ) 2 ] Let ρ X,Y σ X h ( y ) = E [ X ] + ( y E [ Y ]) σ Y = E [ X ] + cov( X, Y ) ( y E [ Y ]) . σ 2 Y b = E [( h ( Y ) X ) 2 ] . (a) Compare the magnitudes of a and b . (b) Suppose all we know about the random variables X and Y is their correlation coeﬃcient Specify all values of ρ X,Y , if any, for which we can be sure that b = 0. Explain ρ X,Y . your reasoning. 2. Problem 4.13 from p.261 of the text. Pat and Nat are dating, and all of their dates are scheduled to start at 9 p.m. Nat always arrives promptly at 9 p.m. Pat is highly disorganized and arrives at a time that is uniformly distributed between 8 p.m. and 10 p.m. Let X be the time in hours between 8 p.m. and the time when Pat arrives. If Pat arrives before 9 p.m., their date will last exacty 3 hours.

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