HW%233_f08_sol

# HW%233_f08_sol - CE 3102 Fall 2008 HW #3 Solutions 1. Let...

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CE 3102 Fall 2008 HW #3 Solutions 1. Let X 1 = 1, if a 40-year flood occurs in the first year 0, if a 40-year flood does not occur in the first year X 2 = 1, if a 40-year flood occurs in the second year 0, if a 40-year flood does not occur in the second year X 40 = 1, if a 40-year flood occurs in the 40 th year 0, if a 40-year flood does not occur in the 40 th year. One way to think about this problem is to note that X 1 , X 2 ,..,X 40 is a Bernoulli sequence (p. 105, A&T), with P(X i = 1) = 1/40 = .025 P(X i = 0) = 0.975 Letting X denote the total number of 40-year floods during the 40-year period, then X is simply = = 40 1 i i X X The sum of independent Bernoulli random variables, with common probability distribution, is a binomial random variable, in this case with parameters n=40 and p=.025. (a) 363 . 0 ) 975 )(. 025 )(. 40 ( ) 025 . 1 ( ) 025 (. )! 1 40 ( ! 1 ! 40 ) 1 ( 39 1 40 1 = = = X P (b) 06 . 0 ) 975 (. ) 025 (. ) 1 )( 2 )( 3 ( ) 38 )( 39 )( 40 ( ) 975 (. ) 025 (. )! 3 40 ( ! 3 ! 40 ) 3 ( 37 3 3 40 3 = = = X P (c ) 637 . 0 . ) 975 (. ) 025 (. )! 0 40 ( ! 0 ! 40 1 ) 0 ( 1 ) 1 ( 40 0 = = = X P X p (d) For this problem, let i=1,. .,20 index the 20 similar systems, and let X i =1, if system i fails at least once during 40 years, 0, otherwise.

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X = = 20 1 i i X Since each X i is a Bernoulli outcome, again X is a binomial random variable. From part (c ) p=P(X i =1) = 0.637, and n=20. An Excel spreadsheet which computes these probabilities is shown below kP ( X = k ) 0 1.57809E-09 1 5.53854E-08 2 9.23319E-07 3 9.72156E-06 4 7.25033E-05 5 0.000407137 6 0.00178613 7 0.006268677 8 0.01787566 9 0.041824777 10 0.080734493 11 0.128795072 12 0.169509217 13 0.183051276 14 0.160611105 15 0.112737492 16 0.06182316 17 0.025526724 18 0.007465805 19 0.001379068 20 0.000121001 (e) X=# failures in 10 years, is binomial with, p=.025, n=10 023 . 0 ) 025 . 1 ( ) 025 (. )! 2 10 ( ! 2 ! 10 ) 2 ( 2 10 2 = = X P 025 . 0 199 . 0 776 . 0 1 )) 1 ( ) 0 ( 1 ( )) 2 ( 1 ( ) 2 ( = = = = < = x P x P X P X P
2. Let X = # of years elapsing until failure. In each year, the probability of failure p=0.10, and the year-to-year flood events are independent, so we have k P ( X = k ) 1, failure in first year p

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HW%233_f08_sol - CE 3102 Fall 2008 HW #3 Solutions 1. Let...

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