Math 115 A - Section 3 - Test 2 Name:___________________29 November 2000In order to get full credit, you must show all of the steps and all of the work that youuse to solve each problem.A list of formulas that you may find useful is provided on the last page of this test.Question Points Score1 122 153 94 355 106 19Total 1001. Let X be the random variable that gives the time in minutes, rounded to the nearestminute, that it will take a randomly selected student to answer this problem.(i) (3 points) Is X a discrete or a continuous random variable ? Explain.(ii) (9 points) Assume that, out of the 26 students in this class, 3 of them willtake 2 minutes to complete this problem, 10 of them will take 4 minutes,and all of the others will take 6 minutes. Use this information to find thefollowing quantities (you may give your answers as fractions):- (p.m.f.) fX (4) = - P( X < 15) =- (c.d.f.) FX (5) =2. In what follows, we consider an investment of $6,000, that you deposit in a bankaccount with a 3.5% interest rate.(i) (5 points) How much will you have in your account after 2 years if interest iscompounded quarterly ?(ii) (5 points) If interest is compounded continuously, what is the yield of your bankaccount ?(iii) (5 points) If interest is compounded monthly, how long should you leave themoney in the bank if you want your balance to be $8,000 ?3. The questions below refer to the following part of an Excel worksheet.A B1 10 0.0034072 11 0.0724753 12 0.9840344 13 0.4101015 14 0.8118766 15 0.643787 16 0.7726888 17 0.7806329 18 0.80195110 19 0.168292(i) (3 points) What is the outcome of the Excel commandVLOOKUP(17,A1:B10,2) ? Explain.(ii) (3 points) Which Excel command would you use to compute the average ofcolumn B?(iii) (3 points) Which Excel command would you use to randomly pick one of thenumbers in Column B ? Explain.4. You are interested in buying a European call stock option that expires in 10 weekswith a strike price of $32. (i) (8 points) Give the definition of a European call stock option. What is theoption worth if the stock closes at $30 on the day the option expires ? What ifthe stock closes at $35 ?(ii) (12 points) Assume that at the end of the 10 weeks, the stock can only close at$25, $30, $35 and $40. Also assume that these closes are equally likely. Whatis the expected value of the stock on the day the option expires ? Show allyour work. (iii) (5 points) Can you use the information given above to find the future value ofyour option ? Why or why not ?(iv) (10 points) Use the information given in Part (ii) to find the possible values ofthe option on the day it expires, together with the corresponding probabilities.Use this information to find the expected future value of the option.5. (10 points) The left two columns of the table shown below were obtained by runningthe "Histogram" function of Excel on a random sample of size n=100. This randomsample consists of independent observations of a continuous random variable X. Usethis information to plot an approximation of the p.d.f. for the random variable X.Think carefully before plotting the graph. Plot the graph on the next page.Bin Frequencyrelative frequencyAdjusted heights3.5 14 24.5 195 425.5 216 126.5 37 0More 0Total 1006. (6 points) Put labels and numbers on the following plot, given that it shows the p.d.f.for a continuous random variable X, which takes values in [0,10]. Then, use thisinformation to answer questions (i) and (ii) on the next page.Probability density function(i) (6 points) What is the value of the p.d.f. fX at x = 8 ? What is the probabilitythat X takes the value 5 ?(ii) (7 points) What is the probability of X being between 4 and 7 ? Explain.FORMULASxuuxu/xx)x(Fxuuxu/x)x(f)xX(Px)X(E ey nr y ePFnrPFXxXrntrnt if 10 if 0 if 0 if 00 if 10 if 0 1 11 1
View Full Document
Unlocking...