2008 5 - Stat 230 Spring 2008 STAT 230 - Spring 2008 - Test...

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Stat 230 Spring 2008 STAT 230 - Spring 2008 - Test 5 SOLUTIONS 1. [14 marks] Four people are listed as friends on Facebook according to the diagram: Friendships ”break” independently of each other, with probability 0.1. Let X i = 1 if person i has no friends 0 otherwise for i = 1 , 2 , 3 , 4 . Let X be the total number of people with no friends. (a) [2 marks] Find P ( X i = 1) for each i . P ( X 1 = 1) = P ( X 2 = 1) = (0 . 1) 2 = 0 . 01 P ( X 3 = 1) = P ( X 4 = 1) = (0 . 1) 3 = 0 . 001 (b) [2 marks] Find E [ X ] using the relationship between X and the X i ’s. X = X 1 + X 2 + X 3 + X 4 So E [ X ] = E [ X 1 + X 2 + X 3 + X 4 ] = E [ X 1 ] + E [ X 2 ] + E [ X 3 ] + E [ X 4 ] = 0 . 01 + 0 . 01 + 0 . 001 + 0 . 001 = 0 . 022 (c) [2 marks] Find Var( X i ) for each i . Var( X 1 ) = Var( X 2 ) = (0 . 01)(1 - 0 . 01) = 0 . 0099 Var( X 3 ) = Var( X 4 ) = (0 . 001)(1 - 0 . 001) = 0 . 000999
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Stat 230 Spring 2008 (d) [5 marks] Find all Cov( X i , X j ), where i < j .
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This note was uploaded on 04/06/2010 for the course STAT 230 taught by Professor Various during the Spring '06 term at Waterloo.

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2008 5 - Stat 230 Spring 2008 STAT 230 - Spring 2008 - Test...

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