1. (a) Suppose that you are standing on the origin of the ry-plane at time t...

 

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1. (a) Suppose that you are standing on the origin of the ry-plane at time t = 0. Each unit time you can move one unit either upward, downward, left or right with equal probability. Therefore at t = 1, you are either at (0, 1), (0, -1), (-1,0) or (1,0) with equal probability. At t = 2, you can be at 9 different positions. Let X, be the distance of you from the origin at time t. Then clearly E(X) = 0 and E(X) 1. It is not difficult to see that = E(X) = (0)+(2)+(2) = 1.2071 Describe in detail a simulation procedure to estimate E(X2020). [8 marks] (b) The inter-arrival time of new customers in a bank is exponentially distributed with a mean of 25 minutes. A new customer must fill in an application form regarding the opening of the new account. It takes between 8 to 12 minutes, uniformly distributed, to complete the task. Then the customer goes to one of two tellers to make the initial deposit. There are separate queues (assume infinite capacity) for each teller and a customer would choose a shorter one. The time required for each teller to complete the deposit is exponentially distributed with a mean of 5 minutes. All customers are served on a "First In, First Out" basis. Describe in detail how a simulation of 12 hours is carried out. [8 marks] 1. (a) Suppose that you are standing on the origin of the ry-plane at time t = 0. Each unit time you can move one unit either upward, downward, left or right with equal probability. Therefore at t = 1, you are either at (0, 1), (0, -1), (-1,0) or (1,0) with equal probability. At t = 2, you can be at 9 different positions. Let X, be the distance of you from the origin at time t. Then clearly E(X) = 0 and E(X) 1. It is not difficult to see that = E(X) = (0)+(2)+(2) = 1.2071 Describe in detail a simulation procedure to estimate E(X2020). [8 marks] (b) The inter-arrival time of new customers in a bank is exponentially distributed with a mean of 25 minutes. A new customer must fill in an application form regarding the opening of the new account. It takes between 8 to 12 minutes, uniformly distributed, to complete the task. Then the customer goes to one of two tellers to make the initial deposit. There are separate queues (assume infinite capacity) for each teller and a customer would choose a shorter one. The time required for each teller to complete the deposit is exponentially distributed with a mean of 5 minutes. All customers are served on a "First In, First Out" basis. Describe in detail how a simulation of 12 hours is carried out. [8 marks]