POISSON PROCESS (total of 36 marks) This workshop is about University of Melbourne students lling up their water bottles at the drinking fountain at the North Entrance of the BBB building. Our experiment consists of observing students lling up their bottles. We assume that we start this experiment at 10am, we call this if = 0. We also assume that students do not visit the drinking fountain twice. Our model involves a number of random processes, including: - T03) = arrival time of the Fc'th student (Fe = 1, 2, . . ) . X (k) = time between the arrival of the k i l'th student and the k'th student (take X (1) = T(1)) . SOC) = time that it takes the k'th student to ll up his/her bottle (=\"service time\") . W06) = waiting time of the Ic'th student . N (t) = the total number of students arriving in the rst t minutes (take N (0) = 0). The interarrival times are assumed to be independent and exponentially distributed with a mean of 4 minutes. Hence, N (t) is a Poisson process with an average arrival rate of A arrivals /minute. You may use the following properties of the Poisson process: 1. The time between two subsequent visits has an ExponentialU.) distribution 2. In any interval of length '7', the number of visits has a PoissonUVr) distribution 3. For any two disjoint intervals, the number of visits in each are independent of each other Questions 1. (1 mark) Based on the given information, what is the numerical value of A for the Poisson process? 2. (3 marks) Suppose that Vi and V2 are independent random variables with V1 ~ Poisson(#1) and V2 ~ Poisson (/2). By using moment generating functions (mof's), show that V1 + V2 ~ Poisson(/1 + (2). 3. a) (3 marks) Find an expression for the second-order CDF of the random process N(t), taking the following into account: - as notation, use time instants t and t2, where 0