Harry\'s boutique is a trendy clothing store on Rodeo Drive inBeverly Hills, Cal

Question

Harry's boutique is a trendy clothing store on Rodeo Drive inBeverly Hills, California. Suppose the arrival rate starts at 0 at 10:00, increases to 4 at 12:00, to 6 by 2:00, drops to 2 by 4:00 and decreases to0 by the time the store closes at 6:00, and that the arrival ratesare linear in between these time points. a) What is the distribution of the number of arrivals in aday? b) What is the probability no one arrives before noon?
Suppose that Harry closes the shop at 5:30 instead of6:00. c) What is the expected number of customers that arelost? d) What is the probability that at least one customer arrivesto find the store closed? Suppose the arrival rate starts at 0 at 10:00, increases to 4 at 12:00, to 6 by 2:00, drops to 2 by 4:00 and decreases to0 by the time the store closes at 6:00, and that the arrival ratesare linear in between these time points. a) What is the distribution of the number of arrivals in aday? b) What is the probability no one arrives before noon?
Suppose that Harry closes the shop at 5:30 instead of6:00. c) What is the expected number of customers that arelost? d) What is the probability that at least one customer arrivesto find the store closed?

Explanation / Answer

2t 0 t 2 t + 2 2 < t 4 -2t +14 4 < t 6 -t + 8 6 < t 8
a) Poisson with mean (t) dt = 24(integral from 0 to 8) b) The distribution of the number of arrivals before noon N isa poisson with mean (t) dt = 4 (integralfrom 0 to 2). So P(N = 0) = e^-4
The distribution of the number of arrivals N during 5:30 -6:00 is Poisson with mean   (t) dt =1/8 (integral from 7.5 to 8)
a) EN = 8 b) P (N > 1) = 1- e^-1/8

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