ECMT5001: In-semester Exam (2022s2)
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ECMT5001：In-semester Exam (2022s2)
Time allowed：1.5 hours
The total score of this exam is 40 marks. Attempt all questions. correct all numerical answers to 2 decimal places.
1. [Total： 15 marks] Bob was studying the traf c low on sydney Harbour Bridge. Let B and C denote, respectively, the number of buses and construction vehicles crossing the bridge in one minute. Equipped with his expertise in buses, Bob decided to model B according to the poisson distribution with mean 4.
(a) [3 marks] what is the probability that more than two buses crossed the bridge in a minute？
(b) [3 marks] Given that more than two buses crossed the bridge in a minute, what is the probability that more than three buses crossed the bridge in a minute？
upon further observation, Bob decided to model C according to the poisson distrib- ution with mean 9. He assumed that the correlation between B and C is 0.6.
The toll company charges $5 for each bus and $20 for each construction vehicle cross- ing the bridge. Let R denote the total revenue generated from buses and construction vehicles in a minute.
(c) [2 marks] compute E(R).
(d) [4 marks] compute Var(R) [Hint：it is true that Var(X) = E(X) if X is poisson distributed.]
(e) [3 marks] Explain to Bob why a poisson model is a poor model to use in practice.
2. [Total： 25 marks] carol has been closely watching the share price of a technology company called BluebeTTY Inc. (with stock code BRRY). Let X denote the change in share price (in number of ticks) in a second (i.e., X = 1 if there is an uptick, = 0 if there is no change, and = -1 if there is a downtick). Based on the historical behaviour of the stock price, carol modelled X according to the following probability density function：
(a) [1 mark] Find E(X).
(b) [2 marks] compute VaT(x).
(c) [3 marks] compute COT(x, x2 ).
In an attempt to validate her model, carol computed the mean price changes of BRRY over n randomly chosen one-second intervals. suppose the-price changes x1, x2 , . . . , xn are iidwith the common distribution given by (*). Let xn = 
x
xi denote the mean price change (in ticks per second).
(d) [2 marks] using the result of part (a), what is E(x(-)2 )？ Justify your answer by
pointing out a property of x2. [Hint：no calculation is needed.]
(e) [2 marks] using the result of part (b), compute VaT(x(-)2 ).
(f) [5 marks] obtain the probability density function ofx(-)2 .
In light of the symmetry of (*) around zero, carol claimed that BRRY breaks even on average (i.e., the mean price change is zero).
(g) [6 marks] Based on a given sample of 80 randomly chosen one-second intervals, there was amean price drop of 0.1 ticks (i.e., 
80 = -0.1). Test at the 5% signiicance level whether carol,s claim is correct. show all your steps. A complete response should include：
i. setting up the null and alternative hypotheses;
ii. deining an appropriate test statistic;
iii. stating the distribution of your test statistic under the null hypothesis; iv. computing the test statistic based on the sampled data;
v. making a decision using a correct method (e.g., critical value approach or p-value approach); and
vi. drawing a conclusion.
(h) [4 marks] By enlarging her sample in part (g) so that n 持 80, carol still observed a mean price drop of 0.1 ticks. will your conclusion in part (g) remain valid or not？ Explain your answer.
2023-08-26