midterm2solutions

midterm2solutions - Math 310-1 Midterm 2 Fall 2011...

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Unformatted text preview: Math 310-1 Midterm 2 Fall 2011 Instructions: Do all work and record all answers in the blue book. No books, notes, or calculators are allowed. You must show ALL of your work, and cross out anything you don’t want graded. Make sure that your final answer is clearly indicated. 1. (8 points): Define the covariance of two random variables X and Y . Solution: Define μ X = E ( X ) and μ Y = E ( Y ). Then the covariance of X and Y is Cov( X,Y ) = E [( X- μ X )( Y- μ Y )] or equivalently, Cov( X,Y ) = E ( XY )- μ X μ Y . 2. (16 points): Let X and Y be two independent random variables, both of which have finite expectations E ( X ) = E ( Y ) = 7 and finite variances Var( X ) = Var( Y ) = 4. (a) Find Var(3 X- Y + 1). (b) Find Cov( X- Y,X + Y ). Solution: (a) Var(3 X- Y + 1) = 3 2 Var( X ) + (- 1) 2 Var( Y ) = 40. (b) Using the definition of covariance, Cov( X- Y, X + Y ) = E [( X- Y )( X + Y )]- E ( X- Y ) E ( X + Y ) = E [ X 2- Y 2 ]- [ E ( X ) 2- E ( Y ) 2 ] = Var( X )- Var( Y ) = 4- 4 = 0 . Page 1 of 4 Math 310-1 Midterm 2 Fall 2011 3. (24 points): Let X be a random variable that has the Poisson distribution with parameter λ ....
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This note was uploaded on 01/03/2012 for the course MATH 310-1 taught by Professor Sarver during the Spring '11 term at Northwestern.

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midterm2solutions - Math 310-1 Midterm 2 Fall 2011...

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