anu sata7055 stat7055_2017-s2_ms-exam_sol

1、Research School of Finance, Actuarial Studies and StatisticsEXAMINATION SOLUTIONSSemester 2 - Mid-Semester, 2017STAT7055 Introductory Statistics for Business and FinanceWriting Time:90 minutesReading Time:15 minutesExam Conditions:Central examination.Students must return the examination paper at the

2、 end of the examination. This examination paper is not available to the ANU Library archives.Materials Permitted in the Exam Venue:(No electronic aids are permitted, e.g., laptops, phones).Calculator (non-programmable).One A4 page with notes on both sides.Unannotated paper-based dictionary (no appro

3、val required).Materials to be Supplied to Students:Script book. Scribble paper.tructions to Students:Please write your student number in the space provided on the front of the script book. Attempt all 5 questions.Start your solution to each question on a new page and clearlylabel each solution with

4、the corresponding question number.To ensure full marks show all the steps in working out your solutions. Marks may be deducted for failure to show working or formulae.Selected statistical tables are attached to the back of the examination paper. Round all numeric answers to 4 decimal places.Question

5、:12345TotalMarksuestion 113 marksDirk likes having coffee and biscuits everyday. Let X denote the number of cups of coffee Dirk drinks on any given day. The number of cups of coffee Dirk drinks is independent from day to day. Let Y denote the number of biscuits Dirk eats on any given da

6、y. The number of biscuits Dirk eats is independent from day to day. The relationship between X and Y is described by the following joint probability distribution:y (Biscuit) 4360.120.250.140.200.100.050.040.020.08367x (Coffee)(a) 3 marks Find the probability that Dirk drank 3 cups of coffee today, g

7、iven that he ate more than 3 biscuits.(b) 3 marks Find the probability that Dirk ate 3 biscuits today, given that he drank more than 3 cups of coffee.(c) 4 marks Over the next 10 days, find the probability that Dirk drank more than 3 cups of coffee on more than 7 days.(d) 3 marks Over the next 30 da

8、ys, find the probability that Dirk drank more than 3 cups of coffee on more than 20 days.Question 213 marksFor this question, use the joint probability distribution of X and Y given in Question 1, where X was the number of cups of coffee Dirk drinks on any given day and Y was the number of biscuits

9、Dirk eats on any given day.Semester 2 - Mid-Semester, 2017Page 2 of 6STAT7055Solution:0.1977Solution:0.1517Solution:0.4068Solution:0.4531(a) 3 marks Find the probability that (X 4)Y 7.Dirk has defined a new quantity, Q = X(Y 4), which he calls his “CC Index” (coffee- cookie index) for that day.(b) 3

10、 marksFind the expected value of Q.4 marks(c)Find the variance of Q.3 marksIf we assume the values of Q are independent from day to day, find the(d)probability that Dirks total cumulative CC Index over the next 60 days is greaterthan 110.Question 312 marksSuppose the number of hours that Dirk works

11、productively on any given day is uniformly distributed between 0 and 2.4 hours. Assume the number of hours that Dirk works productively is independent from day to day.(a) 2 marks If Dirk worked productively for more than an hour today, find the prob- ability that Dirk worked productively for less th

12、an 1.62 hours.(b) 3 marks Over the next 2 days, find the probability that the most amount of time Dirk worked productively on a day was more than 1.62 hours.Semester 2 - Mid-Semester, 2017Page 3 of 6STAT7055Solution:0.5444Solution:0.4429Solution:0.0015Solution:30.93Solution:0.3Solution:0.45(c) 3 mar

13、ks Dirk wants to get a good start to the upcoming 5-day week (Mon - Fri), so on both Monday and Tuesday, Dirk plans to work productively for twice as long as he normally would on any given day. Find the variance of the total number of hours that Dirk works productively over the upcoming 5-day week (

14、Mon - Fri).Dirks excuse for drinking lots of coffee is that it helps him be more productive at work. Suppose now that the number of hours that Dirk works productively on any given day is uniformly distributed between 0 and X hours, where X is the number of cups of coffee Dirk drinks on that day.(d)

15、4 marks Find the probability that Dirk worksproductively for more than 1.62hours today. The joint probability distribution of X and Y given in Question 1 is displayed again below:y (Biscuit) 4360.120.250.040.140.200.020.100.050.08367x (Coffee)Question 49 marksRecall from Question 3 that the numberof

16、 hours that Dirk works productively on anygiven day is uniformly distributed between 0 and 2.4 hours. Suppose that the num-ber of hours that Dirk sleeps on any given day is normally distributed with mean 7.2 hours and standard deviation 1.4 hours. Assume that the number of hours that Dirk works prod

17、uctively and the number of hours that Dirk sleeps, on any given day, are independent.(a) 3 marks Find the probability that Dirk sleeps between 8 and 9 hours today, given that he sleeps for more than 7 hours.(b) 3 marks Find the probability that Dirk sleeps between 7 and 9 hours today or he works pro

18、ductively for more than 0.75 hours today.Semester 2 - Mid-Semester, 2017Page 4 of 6STAT7055Solution:0.3344Solution:0.6282Solution:5.28 hours2(c) 3 marks If W denotes the number of hours Dirk works productively today andS denotes the number of hours Dirk sleeps today, find the variance of 3W + 2S.Que

19、stion 513 marksRecall from Question 4 that the number of hours that Dirk works productively on any given day, denoted W , is uniformly distributed between 0 and 2.4 hours and the number of hours that Dirk sleeps on any given day, denoted S, is normally distributed with mean7.2 hours and standard dev

20、iation 1.4 hours. Suppose now the number of hours that Dirk works productively and the number of hours that Dirk sleeps, on any given day, are not independent.(a) 4 marks If Dirk believes that the variance of 3W + 2S is equal to 7, find the expected value of WS.For parts (b) to (d), assume that P (W 0.75 S 6) = 0.55.(b) 3 marks Find the probability that Dirk sleeps for more than 6 hours today, given that he worked productively for less than 0.75 hours today.(c) 3 marks Find the probability that Dirk sleeps for less than 6 hours today and works productively for less than