Assignment 1 Due 14 Oct, 2021 5pm on Blackboard Write or print your solutions in A4-size paper, convert to one pdf file and submit on blackboard. 1. (20 marks) What are the differences and relationship between Poisson distribution and expo- nential distribution? 2. (20 marks) 36 two-digit decimal numbers as shown below are claimed to be generated from a random process. 53 1 50 99 97 8 17 54 73 88 9 16 67 59 4 92 36 25 89 70 41 43 75 11 49 21 69 83 77 71 91 30 40 81 94 84 Perform run’s test on the claim, with the level of significance at α = 0.05. 3. (20 marks) 36 two-digit decimal numbers as shown below are claimed to be generated from a random process. 53 1 50 99 97 8 17 54 73 88 9 16 67 59 4 92 36 25 89 70 41 43 75 11 49 21 69 83 77 71 91 30 40 81 94 84 Apply the χ2 test on the claim at significance level α = 0.05. Using the intervals: 00-19, 20-39, 40-59, 60-79, 80-99. 4. (20 marks) Apply Kolmogorov-Smirnov’s test on the following data, against uniform distribution on (0,1) at the level of significance α = 0.05. 0.08 0.77 0.54 0.78 0.62 0.30 0.53 0.70 0.59 0.03 5. (20 marks) A one-pump gas station is always open and has two types of customers. A red car arrives every 30 minutes (exactly), with the first red car arriving at time 15 minutes. The distributions of the interarrival time (the time span between two consecutive arrivals) of green cars, and the service time for both kinds of cars are listed below. The first green car arrives at time 0. interarrival time (mins) probability 3 12% 4 15% 5 20% 6 30% 7 23% service time (mins) probability 3 23% 4 30% 5 18% 6 15% 7 14% A car arriving to find the pump idle goes right into service, and green cars arriving to find the pump busy join the end of a single queue. A red car arriving to find the pump busy, however, goes to the front of the line, ahead of any green cars in line. [If there are already other red cars at the front of the line, assume that an arriving red car gets in line ahead of them as well. If a red car and a green car arrive exactly at the same time, the red car is served first. Initially the system is empty and idle, and the simulation is to run until exactly 10 cars (of any type) have completed their delays in queue. Find the average waiting time of the 10 cars. Use the following numbers uniformly distributed on 00–99, to generate the arrival time and service time. Please use the random numbers in a parsimonious way: if there is no need to use a random number, do not use it. When a red car arrives, first make sure the next green car will arrive only after the service of the red car starts, then assign service time for the red car. Namely only assigning service time, when the next event is the service. 56 88 29 79 93 88 56 53 30 66 10 28 53 87 79 61 33 42 9 74 1