a)The number of arrivals N at a supermarket checkout counter in the time interval from 0 to t follows a Poisson distribution with mean λ t . Let T denote the length of time until the first arrival. Find the density function for T. [Hint: P(T>to) = P(N=0at t=to)]
b)The number of arrivals N at a supermarket checkout counter in the time interval from 0 to t follows a Poisson distribution with mean λ t . If U is the time until the second arrival, show that
U has a Gamma density function with α = 2 and β=1/λ
c)Suppose that customers arrive at a checkout counter following a Poisson distribution at a rate of 2 per minute.
1). What are the mean and variance of the waiting times between 2 successive customer arrival?
2) If a clerk takes 3 minutes to serve the first customer arriving at the counter, what is the probability that at least one more customer will be waiting when the service to the first customer is completed?