Assignment 2 - By Mr Terry Lau PDF

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P.STAT S315F Applied Probability Models for InvestmentAssignment 2Due Date: 26th Aug, 2014 (17:00)Full Marks: 55 Two individuals, A and B, both require heart transplants. If she does not receive a new heart, then A will die after an exponential time with rate 0 per month, and B after an exponential ...

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STAT S315F

1.

Applied Probability Models for Investment Assignment 2 Due Date: 26th Aug, 2014 (17:00) Full Marks: 55

Two individuals, A and B, both require heart transplants. If she does not receive a new heart, then A will die after an exponential time with rate 0.5 per month, and B after an exponential time with rate 0.2 per month. New hearts arrive in accordance with a Poisson process having rate 2 per month. It has been decided that the first heart will go to A (or to B if B is alive and A is not at that time) and the next one to B (if still living). (a) What is the probability that A obtains a new heart? (b) What is the probability that B obtains a new heart?

2.

Let {N(t), t ≥ 0} be a Poisson process with rate λ. Let Sn denote the time of the nth event. Find (a) E[S5], [1M] (b) E[S6|N(2) = 3], (c) E[N(10) − N(6)|N(2) = 3].

3.

[2M] [4M]

[2M] [2M]

A store opens at 9 A.M. From 9 until 11 A.M. customers arrive at a Poisson rate of three an hour. Between 11 A.M. and 1 P.M. they arrive at a Poisson rate of six an hour. From 1 P.M. to 3 P.M. the arrival rate increases steadily from six per hour at 1 P.M. to eight per hour at 3 P.M.; and from 3 to 6 P.M. the arrival rate drops steadily from eight per hour at 3 P.M. to two per hour at 6 P.M.. (a) Find the mean and variance of the number of customers that enter the store on a given day. [5M] (b) Find the probability that there are more than 16 but less than 20 customers enter the store from 2 to 4 P.M. on a given day.

[4M]

P.1

4.

Members of the public arrive independently and at random at an advice bureau where there are five advisers on duty. If an adviser is free, then a person entering the bureau receives immediate attention. Otherwise, a central queue is formed. The time that an adviser spends with a member of the public is exponentially distributed with mean ten minutes and is the same for all the advisers. On average, three members of the public arrive every ten minutes. (a) Write down the specification of this queue, and calculate the traffic intensity ρ. Assume that the queue is in equilibrium. (b) Calculate the proportion of the time that all five advisers are free. (c) What proportion of the time are exactly three advisers busy?

[2M] [3M] [3M]

(d) Calculate the probability that when I arrive at the bureau, all the advisers will be busy and one person will be waiting for advice. [3M] (e) Sam arrives to find that all five advisers are busy but no one is waiting for advice. Calculate the probability that he will have to wait for at least five minutes for an adviser to be free to attend to him.

5.

[3M]

A child appears at the front door of her home, which is one of a very long terrace. The child’s play consists of trotting up and down along the pavement, but there is no pattern apparent to her play. It is decided to model the random process {X(t); t ≥ 0} denoting her displacement from her front door at time t by ordinary Brownian motion with diffusion coefficient σ2 = 180 m2 per hour. (a) Calculate the probability that after half an hour’s play the child is more than 15 metres away from her front door. [4M] The child leaves her front door to play on the pavement one morning at exactly 10 o’clock, and is seen outside her front door (not necessarily for the first time) at 10:40. (b) How would you model the child’s distance from home between 10:00 and 10:40? [2M] (c) Calculate the probability that at 10:20 the child was less than 10 metres from home. [5M] (d) Calculate the probability that at 10:30 the child was less than 10 metres from home. [5M] (e) Calculate the probability that at 11 o’clock the child will be more than 15 metres away from home.

[5M] End

P.2...