Math

Starting at time t=0, cars arrive at a car wash according to a Poisson process with rate of λ cars/hour. At any given moment, the car wash is either free or occupied. The car wash is initially free at time t=0. If a car arrives at the car wash when it is free, the car is serviced immediately. Service lasts for 1/4 hours, during which the car wash is occupied. If a car arrives at the car wash when it is occupied, the car is denied service and it leaves the car wash.

Question 1--> Write down the PMF pN(k) of N, the number of cars arriving at the car wash between times 0 and 3, in terms of λ and k.

For k=0,1,…,

Question 2--> Find the probability that a car is accepted for service given that it arrives at time t=1/6. Write down your answer in terms of λ, using the standard notation.

Question 3--> Find the PDF fT2(t) of T2, the time until the second serviced car leaves the car wash.
For t≥0.5,fT2(t) =?

Question 4--> Find the probability that exactly one car is accepted for service between t=0 and t=1.

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