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final_2007

final_2007 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Fall...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Fall 2007 6.436J/15.085J± Final exam, 1:30–4:30pm, (180 mins/100 pts) 12/19/07± Problem 1: (24 points) During the time interval [0 ,t ] , men and women arrive according to independent Poisson processes with parameters λ 1 and λ 2 , respectively. With the exception of part (e), just provide answers (possibly based on your intuitive understanding)—justifications are not required. (a) (3 pts.) Let [ a,b ] be an interval contained in [0 ] . Give a formula for the prob- ability that the total number of male arrivals during the interval [ ] is equal to 7. (b) Out of all the people who arrived during [0 ] , we select one at random, with each one being equally likely to be selected. (i) (3 pts.) Write an expression for the probability that the selected person is male. (ii) (3 pts.) Suppose that the randomly selected person tells us that he/she ar- rived at a particular time τ . What is the conditional probability that this person is male? (iii) (3 pts.) Write an expression (as simple as you can) for the expected time at which the selected person arrived. (c) (4 pts.) Suppose that 0 < a < b < t . Let N 1 be the number of male arrivals during [0 ,b ] . Let N 2 be the number of female arrivals during [ a,t ] . What is the probability mass function of N 1 + N 2 ? (d) (4 pts.) Suppose that in (c) above we are told that N 1 + N 2 = 10 . Find the conditional variance of N 1 , given this information. (e) (4 pts.) Find a good approximation for the probability of the event { the number of arriving men during [0 ] is at least λ 1 t } , when t is large, and justify the approximation. Solution: (a) e λ 1 ( b a ) ( λ 1 ( b a )) 7 7! (b)(i) λ 1 / ( λ 1 + λ 2 ) (ii) Since the time a person has arrived is independent of whether he was classified into male or female, the answer is the same as in (i). 1
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(iii) The distribution of a randomly selected arrival is uniform over [0 ,t ] . Its expectation is t/ 2 . (c) Male and female arrivals are independent processes, and the sum of independent Poisson random variables is again Poisson. The answer is Poisson with parameter λ 1 b + λ 2 ( t a ) . (d) Since each arrival N 1 + N 2 independently comes from N 1 with probability λ 1 b p = , λ 1 b + λ 2 ( t a ) the distribution of N 1 conditional on N 1 + N 2 = 10 is binomial with parameters n = 10 and p . Its variance is 10 p (1 p ) .
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final_2007 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Fall...

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