ECMT5001: In-semester Exam (2023s1)
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ECMT5001: In-semester Exam (2023s1)
Time allowed: 1 hour
April 2023
The total score of this exam is 40 marks. Attempt all questions.
The following information will be useful for answering the next 6 questions.
Carol has been closely watching the share price of two technology companies: Avocado Inc. (with stock code AVO) and Blueberry Inc. (with stock code BRRY). Let A and B respectively denote the change in the share price of AVO and BRRY (in number of ticks) in a second (i.e., = 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 the price movement of AVO and BRRY according to the following joint probability density function:
|
|
B = _1 |
B = 0 |
B = 1 |
|
A = _1 |
0:23 |
0:05 |
0:02 |
|
A = 0 |
0:05 |
0:5 |
0:05 |
|
A = 1 |
0:02 |
0:05 |
0:03 |
Q1: The value of E(A) is
Q2: The value of Var(A) is
Q3a: The marginal probability density function of B is fB (b) =?
Q3b: The marginal probability density function of A is fA (a) =?
Q4a: The conditional probability density function of B given A = 1 is fB|A(b|1) =? Q4b: The conditional probability density function of A given B = 1 is fA|B(a|1) =?
Q5a: The correlation of A and B is
Q5b: The covariance of A and B is
Q6: Are A and B independent?
The following information will be useful for answering the next 7 questions.
Mimi and Baobao are Simonís dog and cat. They visit Simonís room from time to time. The number of animals X appearing in Simonís room at any given time follows the probability distribution below:
|
x |
0 |
1 |
2 |
|
fX (x) |
0:2 |
0:3 |
0:5 |
Let 
n and sn(2) denote respectively the expected value and variance of the number of animals in Simonís room at randomly chosen time instants.
Q7: Let g(.) be the probability density function of 
2 . Compute the values of g(x) at x = 0; 0:5; 1; 1:5; 2.
Q8: Let h(.) be the probability density function of s2(2) . Compute the following values of h(x) at x = 0; 0:5; 1; 1:5; 2.
Q9: The expected value of 
10 is:
Q10: The variance of 
10 is (give exact answer):
Q11: As the sample size n increases, what does the sample mean 
n converge in probability to? Q12: As the sample size n increases, what does P(
n > E(X)) converge to?
Q13: Carol obtained a random sample of 100 time instants and computed the sample mean 
100 . What is the probability that 
100 is within the true mean by less than 0.1? Correct your answer to 2 decimal places.
The following information will be useful for answering the next 6 questions.
A phone company claims that their phone battery can last more than a day (24 hours) without charging. To test this claim, Robert collected a random sample of 25 phones and recorded how much time each phone can last without charging. The sample mean is 25 hours and the sample standard deviation is 2.5 hours.
Q14: What are the null and alternative hypotheses?
Q15: What is the distribution of the test statistic under the null hypothesis?
Q16: Find the p-value. Correct your answer to 3 decimal places.
Q17: What is the conclusion of the test at the 1/3/5% signiÖcance level?
Q18: Robert decided to increase the sample size to 100/increase the signiÖcance level of the test. What would be the e§ect on the test power?
Q19: Robert decided to increase the sample size to 100/increase the signiÖcance level of the test. What would be the e§ect on the probability of type II error?
2023-06-06