ECMT5001: In-semester Exam (2022s1)
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ECMT5001: In-semester Exam (2022s1)
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 Bob,s burger, which is priced at $10 each.
The number of Bob,s 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 proit.
(b) [1 mark] Express P in terms of C and N.
(c) [2 marks] compute E(P).
(d) [3 marks] compute Var(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 > 2|y > 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 signiicance level. 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.
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.
(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, deined below
Let |X| denote the absolute value of X (e.g., |-2| = 2, |2| = 2). Note that |X| = SX and X = S |X|.
i. [3 marks] compute cor(S, |X|), the correlation between S and |X|. 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?
2023-08-26