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ECMT5001: In-semester Exam
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: 9 marks] Bob is the proud owner of the restaurant Hungry Bob. The only
product Hungry Bob sells is Bobs burger, which is priced at $10 each. The number of
Bobs burgers sold on a day, denoted N , follows a normal distribution with mean 400 and
standard deviation 50.
(a) [3 marks] What is the probability that the daily revenue exceeds $5,000?
It is known that the total daily cost, denoted C, follows a normal distribution with
mean $1,000 and standard deviation $300. The correlation between C and N is 0.8.
Let P denote the total daily pro
t.
(b) [1 mark] Express P in terms of C and N .
(c) [2 marks] Compute E(P ).
(d) [3 marks] Compute V ar(P ).
2. [Total: 16 marks] The government reported that the infection rate of COVID-19 is 0.2.
Let Y denote the number of people infected with COVID-19 in a random sample of 5
individuals.
(a) [3 marks] What is the distribution of Y ? Name the distribution and specify its
parameter(s).
(b) [3 marks] Compute P (Y > 1).
(c) [4 marks] Compute P (Y > 2jY > 1).
(d) [6 marks] Simon wanted to test whether the true infection rate is higher than 0.2.
He collected a random sample of 100 individuals. It was found that 27 individuals
were infected with COVID-19. Carry out a hypothesis test for Simon at the 5 percent
signi
cance level. Show all your steps. A complete response should include:
i. setting up the null and alternative hypotheses;
ii. de
ning 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.
1
3. [Total: 15 marks] Carol is a trader for an investment bank in Wall Street. She is studying
the tick movement of a blue chip stock. Let X denote the price change (in number of ticks).
The probability density function of X is given below.
x f(x)
-2 0.02
-1 0.08
0 0.8
1 0.08
2 0.02
(a) Compute the following:
i. [2 marks] E(X)
ii. [2 marks] E(X2)
iii. [2 marks] sd(X)
(b) Let S denote the sign of X, de
ned below
S =
8<:
1 if X is negative,
0 if X = 0,
1 if X is positive.
Let jXj denote the absolute value of X (e.g., j2j = 2, j2j = 2). Note that jXj = SX
and X = S jXj.
i. [3 marks] Compute Cor(S; jXj), the correlation between S and jXj.
ii. [4 marks] Compute Cor(S;X), the correlation between S and X.
iii. [2 marks] Are S and X independent? Why or why not?