Nowadays, movies can be rented from a vending machine located at the entrance to
ID: 3220320 • Letter: N
Question
Nowadays, movies can be rented from a vending machine located at the entrance to many stores. Suppose that it is now Friday evening at 6pm and a certain machine within a certain store has seven copies of the movie "Twilight" available for rent. The machine will not be visited by the owner until Monday morning at 6am (which is 60hrs later), at which time returned movies will be restocked. Suppose that customers wanting to rent "Twilight" arrive at this rental machine at a rate of 1 every 8 hours. Let X = the number of renters wanting "Twilight" that come to the vending machine over the weekend (Fri 6pm until Mon 6am). a) Name the distribution of X and identify the parameter. Distribution name: ________ Parameter value: __________ b) What is the probability that exactly 5 copies of "Twilight" are rented over the weekend? Thus, we seek the P (X = 5) which equals _______? c) What is the probability that all copies of "Twilight" are rented over the weekend? Thus, we see P(X __ __) which equals ________?Explanation / Answer
Distribution Name: Poisson distribution, number of events occurring in a fixed interval of time.
Parameter Value: (1/8)*60 = 7.5 every 60 hours - The rate parameter = 7.5 for 60 hours
PDF of Poisson is (ke-)/k!, P(X=5) = (7.55*e-7.5)/5! = 0.10937
For sellout, we need atleast 7 renters to show up, therefore P(X>=7) = 1 - (P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)) = 1-((7.50*e-7.5)/0! + (7.51*e-7.5)/1! + (7.52*e-7.5)/2! + (7.53*e-7.5)/3! + (7.54*e-7.5)/4! + (7.55*e-7.5)/5! + (7.56*e-7.5)/6!) = 0.62185
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