ECMT5001: In-semester Exam (2022s2)
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
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 tra¢ c áow 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 Blueberry 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 = 一1 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 Var(X).
(c) [3 marks] Compute Cor(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 ;:::;Xnare iid with the common distribution given by (*). Let 
n = 
x
Xi denote the mean price change (in ticks per second).
(d) [2 marks] Using the result of part (a), what is E(
2 )? Justify your answer by pointing out a property of 
2 . [Hint: no calculation is needed.]
(e) [2 marks] Using the result of part (b), compute Var(
2 ).
(f) [5 marks] Obtain the probability density function of 
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 a mean price drop of 0.1 ticks (i.e., 
80 = 一0:1). Test at the 5% signiÖcance level whether Carolís claim is correct. Show all your steps. A complete response should include:
i. setting up the null and alternative hypotheses;
ii. deÖning 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 > 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.
2023-06-06