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# STAT135_9 - Stat 135 Fall 2006 HOMEWORK 9(due Friday 11/17...

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Stat 135, Fall 2006 HOMEWORK 9 (due Friday 11/17) You have two weeks to do these problems. If you decide to take a week off and just do them in the second week, you will not find me helpful. On Tuesday 11/14 I will help you with at most three problems, and on Thursday 11/16, I will help you with at most one part of one problem. 1. 12.26. 2. 12.27. 3. 12.28. 4. Let X and Y be two square-integrable random variables. That means E ( X 2 ) and E ( Y 2 ) are both finite. Define the correlation between X and Y to be R ( X, Y ) = E X - E ( X ) SD ( X ) Y - E ( Y ) SD ( Y ) Let X * be X in standard units, so that X * = ( X - E ( X )) /SD ( X ). Let Y * be Y in standard units. a) Show that R ( X, Y ) = R ( Y, X ) = R ( X * , Y * ). This means that the correlation between two variables does not depend on the order of the variables nor on the units in which the variables were measured. b) Show that - 1 R ( X, Y ) 1. [Hint: The random variables ( X * + Y * ) 2 and ( X * - Y * ) 2 are non-negative, therefore so are their expectations.] In what follows, you have two observations on each of n individuals. The observations are ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ). Define ¯

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STAT135_9 - Stat 135 Fall 2006 HOMEWORK 9(due Friday 11/17...

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