461s11t1s - Math 461 Test 1, Spring 2011 Calculators,...

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Unformatted text preview: Math 461 Test 1, Spring 2011 Calculators, books, notes and extra papers are not allowed on this test! Show all work to qualify for full credits 1. (16 points) (a) Suppose that A 1 ,A 2 ,A 3 ,A 4 ,A 5 are independent. Given that P ( A 1 ) = P ( A 2 ) = 1 2 , P ( A 3 ) = P ( A 4 ) = 1 3 , P ( A 5 ) = 1 4 . Find P (( A 1 A 2 ) ( A 3 A 4 ) A 5 ). (b) Suppose X is a geometric random variable with parameter p = 2 3 , find E [(1- X ) 2 ]. Solution (a) P (( A 1 A 2 ) ( A 3 A 4 ) A 5 ) = P ( A 1 A 2 ) P ( A 3 A 4 ) P ( A 5 ) = ( P ( A 1 ) + P ( A 2 )- P ( A 1 A 2 ))( P ( A 3 ) + P ( A 4 )- P ( A 3 A 4 )) P ( A 5 ) = ( 1 2 + 1 2- 1 4 )( 1 3 + 1 3- 1 9 ) 1 4 = 5 48 . (b) E [(1- X ) 2 ] = E [1- 2 X + X 2 ] = 1- 2 E [ X ] + E [ X 2 ] = 1- 2 E [ X ] + (Var( X ) + ( E [ X ]) 2 ) = 1- 2 3 2 + ( 3 4 + 9 4 ) = 1 . 2. (13 points) a 9-card hand is randomly selected from an ordinary deck of 52 cards without replacement. Find the probability that the hand contains all 4 cards of at least 1 of the 13 denominations....
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This note was uploaded on 04/03/2012 for the course STAT 461 taught by Professor Renmingsong during the Fall '11 term at University of Illinois, Urbana Champaign.

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461s11t1s - Math 461 Test 1, Spring 2011 Calculators,...

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