Exam1Bans - Exam 1 1. (Version B yellow) (Answers) Suppose...

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Exam 1 ( Version B – yellow ) ( Answers ) 1. Suppose the moment-generating function of a random variable X is ( 29 t t e e t 5 2 X 1 . 0 6 . 0 3 . 0 M + + = . a) (3) Find the probability P ( X 3 ). x f ( x ) 0 0.3 2 0.6 5 0.1 P ( X 3 ) = f ( 0 ) + f ( 2 ) = 0.3 + 0.6 = 0.9 . b) (3) Find the expected value of X, E ( X ). M X ' ( t ) = t t e e 5 2 5 . 0 2 . 1 + . E ( Y ) = M X ' ( 0 ) = 1.2 + 0.7 = 1.7 . OR x f ( x ) x f ( x ) 0 0.3 0.0 2 0.6 1.2 5 0.1 0.5 1.7 E ( X ) = x x f x all ) ( = 1.7 . c) (4) Find the variance of X, Var ( X ). M X '' ( t ) = t t e e 5 2 5 . 2 4 . 2 + . E ( X 2 ) = M X '' ( 0 ) = 2.4 + 2.5 = 4.9. OR x f ( x ) x 2 f ( x ) 0 0.3 0.0 2 0.6 2.4 5 0.1 2.5 4.9 = E ( X 2 ). Var ( X ) = E ( X 2 ) – [ E ( X ) ] 2 = 4.9 – 1.7 2 = 2.01 .
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2. Alex sets two alarm clocks each night to ensure that he does not sleep through his 9:00 a.m. class. His primary clock properly sounds its alarm on 80% of the mornings, while his secondary clock rings its bell on only 70% of mornings. Assume the clocks operate independently. a) (3) What percent of the time does Alex’s two-clock strategy prevent him from oversleeping? That is, find the probability that at least one alarm would sound on a given morning. P( 1st OR 2nd ) = P( 1st ) + P( 2nd ) – P( 1st AND 2nd ) = 0.80 + 0.70 – 0.80 × 0.70 = 0.94 . OR P( at least one ) = 1 – P( none ) = 1 – 0.20 × 0.30 = 0.94 . b) (3) Find the probability that only one alarm would sound on a given morning. P( 1st only ) + P( 2nd only ) = 0.80 × 0.30 + 0.20 × 0.70 = 0.38 . 3. (3) Suppose an individual is randomly selected from the population of adult males living in the United States. Let A be the event that the selected individual is over 6 ft. in height, and let B be the event that the selected individual is a professional basketball player. Which do you think is larger, P( A | B ) or P( B | A ), and why ? ( Circle one and
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This note was uploaded on 04/27/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Exam1Bans - Exam 1 1. (Version B yellow) (Answers) Suppose...

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