OR-Midterm(2006-1)_solution.doc - 연세대학교 정보산업공학과 OR Stochastic Models Midterm Exam(2006 ID Name 1(35pts A and B play a series of

# OR-Midterm(2006-1)_solution.doc - 연세대학교...

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연세대학교 정보산업공학과 OR Stochastic Models, Midterm Exam.(2006) April 20, 2006 ID:____________________ Name:____________________ 1. (35pts) A and B play a series of games with A winning each game with probability p . The overall winner is the first player to have won two more games than the other. (a) Find the probability that A is the overall winner. (20pts) (b) Find the expected number of games played. (15pts) Sol) a) – i) 2 2 2 2 2 1 ) 1 ( 2 1 ) ( , ) 1 ( 2 ) ( 0 | ( ) 0 ( ) 0 | ( ) ( # : # : p p p p p A P Thus p p p A P Y A P Y P Y A P A P two first the in A for wins of Y played games of X over the is A that event the be A Let ii) 2 2 2 2 2 2 1 ) ( , ) ( ) ( ) ( ) 1 ( ) 1 )}( ( { )} ( ) 1 ( { ) 0 ( ) 0 | ( ) 1 ( ) 1 | ( ) ( , 0 , 1 p p p A P Thus A P p A pP A P p p p p A pP p A P p p I P I A P I P I A P A P game first the in loser the is A game first the in winner the is A I b) – i)

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2 2 2 2 2 1 2 ) 1 ( 2 1 2 ] [ , ) 1 ( 2 ] [ 2 2 ) 1 ( 2 ]) [ 2 ( ) 1 ( 2 ) 2 ( ] 2 | [ ) 1 ( ] 1 | [ ) 0 ( ] 0 | [ ]] | [ [ ] [ p p p p X E Thus p p X E p p p X E p Y P Y X E Y P Y X E Y P Y X E Y X E E X E 2. (30pts) Customers arrive to a bank according to a Poisson process of rate λ. Suppose three customers arrived in the first hour. What is the probability that one customer arrived in the first 15 minutes?
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