INFORMATION ABOUT SMAM

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1 INFORMATION ABOUT SMAM 351 Course Title: Probability and Statistics Textbook: PROBABILITY AND STATISTICS FOR ENGINEERING AND THE SCIENCES Sixth Edition by Jay L. Devore Duxbury Press Course Content: An introduction to basic concepts of Probability and Statistics. The bulk of Chapters 1 to 5. Instructor: Dr.M.Gruber.Office 8-35.Phone e mail mjgsma@rit.edu There is a web page for the course located at the address The handout passed out on the first day has the homework for the first week. The rest of the assignments are on the web page. Homework: See assignment sheet. Homework will not be collected. Some of it will be gone over in class. There will be minute homework quizzes and three short computer assignments each counting as a quiz. The homework quizzes will consist of part of or rone or two of the homework problems. The homework quiz grade will be the average of the best 8 quizzes. Attendance: You are strongly advised to attend every class except for serious illness, death in the immediate family, religious holidays and being sent out of town by an employer. Attendance will be taken randomly on about 5 or 6 nights during the quarter. Students that are there on one of these nights will have a point added to their homework quiz grade. Examinations: There will be 3 one-hour examinations. There will be a two-hour final as scheduled by the insititute during final exam week. The last final exam is Friday May 6,6. Until the exam schedule comes out keep that in mind when you make travel plans. There will be no early exams. Dates for hour exams are given on the bottom of this page. Grades: Each hour test counts %. Homework quizzes count %. The final exam counts %. The final exam grade can replace two of the above mentioned items if it is better, e.g., one or two hour exams, the homework quizzes and an hour exam and thus count 4% or 6% of the grade. Unexplained absence from the final exam results in an automatic "F. A missed hour test or quiz or a computer assignment not turned in regardless of the reason receives a zero. Warning: In deciding borderline cases important factors include whether all of the tests and quizzes were taken. Comment: The examination and quiz schedule is set up so that you get feedback on how you are doing on a weekly basis. Take advantage of it by taking all of the tests and quizzes.. Plan to attend every class and at least try all of the problems on the assignment sheet. If you have time do more of the problems in the text. Remember, Mathematics is not a spectator sport; practice makes perfect. I am here to help you learn the material but you have to meet me halfway. I will generally be available in my office and in tutoring. Graded Homework Instructions Exam and Quiz Dates Exams March 3, April 18, May 11 Quizzes March 1 March 3 April 6 April 13 April 7 May 4 May 18 Computer Assignments due April,April 7, May 18 Availability Office 8-35 MTWR 1-PM Bates Study Center -3 MW 4:3-5:3 TR SMAM 351 Assignments Spring 6 Dr. Gruber Chapter.1: 57:, 3,4, 5, 8. Suppose the sample space S = {1,,3,4,5,6,7,8,9}, A={1,3,5,7}, B={6,7,8,9}, C={,4,8} and D = {1,5,9}. A. List the elements of the subsets of S that correspond to the following events

2 (1) A B () ( A B) C (3) B C (4) ( B C) D (5) A C (6) ( A C) C B. Suppose it is given that P(A) = 4 9,P(B) = 4 9,P(C) = 3 9,P(D) = 3 9,P(A B) = 1,P(A C) =, 9 P(A D) = 1 9,P(B C) = 1 9,P(B D) = 1 9 Intersections of three or four of the subsets A, B, C, D have probability zero.(why?) Find the probability of each of the subsets in part A. C. Give a physical problem that will lead to the probabilities in part B.: 64: 11,1,13, 14,15, 17,18, 19,,1,, 5. A biology professor has two graduate assistants helping her with her research. The probability that the older of the two assistants will be absent on any given day is.8 the probability that the younger of the two will be absent on any given day is.5 and the probability that they will both be absent on any given day is.. Find the probabilities that A. at least one of the graduate assistants will be absent on any given day; B. at least one of the graduate assistants will not be absent on any given day; C. only one of the two graduate assistants will be absent on any given day..3: 73: 9,3,31, 3,33,34, 36,37,4, 43. A carton of 15 light bulbs contain one that is defective. In how many ways can the inspector select 3 of the bulbs and A. get the one that is defective? B. get the one that is not defective?.4 :83: 45, 47, 5, 53, 55,57,58 59,6,61, 6,63, 65 It is well known from experience that in a certain industry 6% of all labormanagement disputes are over wages, 15 percent are over working conditions, and 5% are over fringe issues. Also 45% of the disputes over wages are resolved without strikes, 7 percent of the disputes over working conditions are resolved without strikes, and 4 percent of the disputes over fringe issues are resolved without strikes. A. What is the probability that a labor management dispute will be resolved without a strike? B. Assume a labor management dispute was resolved without a strike. What is the probability it was over wages?.5: 9: 69,7, 73, 74, 75, 76, 77,78,79,81, 8, 83 A system consists of two subsystems connected in series. Subsystem 1 consists of two components with reliability.99 and.98 connected in series. Subsystem consists of five components connected in parallel with reliabilities.75,.6,.65,.7 and.6. Find the system reliability. Chapter 3 3.1: 1: 1, 7, 8,1 3.: 19: 11,13,14,15, 16, 17,1,,3, 7

3 Suppose that random variable X has the distribution function x < x < 1 F(x) =.5 1 x < x < 5 1 x 5 A. Find each of the following probabilities (1)P(X 3) ()P(X = 3) (3)P(X < 3) (4)P(X 1) (5)P(.4 < X < 4) (6)P(X = 5) B. Find the probability mass function of random variable X. 3.3: 118: 8,9,3,31,3, 33, 34, 35, 36, 38, 41 The probability distribution X of the number of accidents each week at a certain intersection is given by x 1 3 p(x) A. What is the probability that the number of accidents in a given week is at least? B. What is the mean number of accidents per week? C. What is the probability that the number of accidents in a week is at most (1) one standard deviation from the mean? () at most 1.5 standard deviations from the mean? (3) less than two standard deviations from the mean? D. Find the cumulative distribution function of X. E.Suppose that the number if accidents in week 1 and week are independent. In each of the two weeks the above probability distribution is followed.what is the probability that the total number of accidents during the two-week period is exactly four? 3.4: 16: 45,47,48,49,5, 51, 53, 57, 59, 61 An automobile safety engineer claims that one in ten automobile accidents are due to driver fatigue. For twenty automobile accidents what is the probability A. At most three are due to driver fatigue? B. Between two and five accidents inclusive are due to driver fatigue? C. At least 15 accidents are due to a cause other than driver fatigue? [Hint:Use the binomial table] 3.5: 133: 65,67, 69, 7, 71,7 1.The probability is.75 that a student will believe a rumor about his/her professor s grading policies. Suppose students are asked independently whether they believe the rumor? A. What is the probability that at most five students must be interviewed until one says he believes the rumor? B.What is the probability that exactly six students must be interviewed until three who believes the rumor are found?\ 3.6: 138: 75,77,78,79, 81,8,83,84,87,88, 89 3

4 Chapter 4 4.1: 15: 1,, 3, 4,5,6, 7,8, 1 4.: 159: 11,13,15,18,,1,3,4, 5 1.A random variable X has the probability density function x < x 1 1 1< x f (x) = 3 x < x < 3 elsewhere Find P(.5<X<.5) B. Find the cumulative distribution function of X. C. Find the mean of X. The proportion of people who respond to respond to a certain mail order solicitation is a continuous random variable that has the density function (x + ) < x <1 f (x) = 5 elsewhere A. Find the probability that more than 1/4 but fewer than 1/ of the people contacted will respond to this type of solicitation. B. Find the probability that at least /3 of the people contacted will respond to this kind of solicitation. C. Find the probability that the proportion of people who will respond to this type of solicitation is within 1.5 standard deviations of the mean. D. Compare your answer to part C to the lower bound obtained using Chebychev s Theorem. 3. The time to failure of a machine component in thousands of hours has CDF F(x) = 1 e x/5,x > A. What is the probability the part still functions after 75 hours. B. What is the expected life of the part? Problems on Chebychev s Inequality See handout on Web page. In addition Do 119:43, 17:63 and the first problem below : 6, 7,9, 3, 31,35,37,39,41, 43,47,48,49,5,51 1.Let X be any random variable with mean 5 and standard deviation.5. A. Find a lower bound on P(4 < X < 58) using Chebychev s Theorem B.Find an upper bound on (1) P(X> 6)+P(X< 4) () P(X< 4) C. Find the value of the constant c where 4

5 P( X 5 c).9 D. Assuming X is normally distributed with mean 5 and standard deviation.5 find the exact probabilities for parts A and B.. In the November 199 issue of Chemical Engineering Progress a study discussed the percent purity of oxygen from a certain supplier. Assume that the mean was with standard deviation of.8. Assume that the distribution of percent purity was approximately normal. A. What percentage of the purity values would you expect to be between 99.5 and 99.7? B..What purity value would you expect exactly 5% of the population to exceed? Use the normal approximation to the binomial distribution to do this problem. 3. A commonly used practice of airline companies is to sell more tickets than actual seats on a particular flight because customers who buy tickets do not always show up for the flight. Suppose the percentage of no shows at flight time is %. For a particular flight with 197 seats, a total of tickets were sold. What is the probability that the airline overbooked the flight? 4.4: 179: 55,56,57, 58, 59, 61,63, The time until an appliance needs repair has an exponential distribution with mean 4, hours. A. What is the probability that the appliance will need repair after at most 3, hours? B. Assuming the appliance has not needed repair for 3, hours what is the probability it will need repair during the next 3, hours?. The proportion of a brand of a television set that requires repair during the first year is a random variable having a beta distribution with α = 3,β =. What is the probability that at least 8% of the new models sold this year will require repairs during the first year of operation? 4.5: 187: 66,67,69, 75,78,79 1.The service life of a semiconductor is a random variable having a Weiibull distribution with α =.5,β = 16. A. How long can such a semiconductor be expected to last? B. What is the probability the semiconductor is still working after 4hours? Chapter 5 Review of Multiple Integrals (See over) : Discrete Random Variables 15 and 4: 1,3,4,7,1, 13, 15, 17,,3,3 7, 3 Continuous Random Variables 1.Let X and Y be continuous random variables with joint density function (x + y) x 1, y 1 3 f (x,y) = elsewhere A. Find the marginal density of X and of Y. B.Find P(X <.5) C. Find P(X<.5 Y=.5) D.Find cov(x,y) and the correlation coefficient. 1. The continuous joint density function of the random variables X and Y is 5

6 f (x,y) = 6x, < x <1, < y <1 x elsewhere A. Determine whether random variables X and Y are independent. B. Find P(X>.3 Y=.5) C. Find Var(X Y) 15: 1,13 1:7 5.4: 4: 48,49,5,53, 55,57 5.5: 46: 59, 63,64, 65,69,73, Chapter 1 3:1,15,16 (For each Data set make a five number summary find x and s. Determine whether there are any outliers and comment on the shape of the data.) Computer Assignment 1 due 4//6 1.Use Minitab to do this problem. The probability that a car stolen in a certain western city will be recovered is.65. Out of 4 cars what is the probability A. At least 5 are recovered? B. At most 1 are not recovered?. Use Minitab to do this problem The probability is.15 that a person will get food poisoning spending the day at a certain state fair. A. Use the binomial distribution to find the exact probability that at most two people out of 1 who attend the fair will get food poisoning. B. Use the Poisson approximation to approximate the answer to A. Comment on the approximation. Computer Assignment Due 4/7/6 1.. Do this problem using the software two ways. First use the normal approximation to the binomial distribution with the continuity and then without the continuity correction. Then calculate the exact binomial probabilities. Compare the results and tell which of the two approximations is better. A random sample of 5 batteries is selected from a very large shipment of battery cases that is 5% nonconforming. What is the probability that there at least nonconforming batteries?. The lifetime in weeks of a certain type of transistor follows a gamma distribution with mean 18 weeks and variance 18 weeks. A. What is the probability the transistor will last at most 1 weeks? B. What is the median time to failure of the transistor? Computer Assignment 3 Due 5/17.Use Minitab to do this problem that demonstrates The Central Limit Theorem Put the numbers, 1,,3,4 into column 1. Put the first five digits of your Social Security Number in c. Enable the command language. At the prompt put in the command 6

7 let c3 = c/sum(c) Now do the following computations and show your results on the page stapled to the front of the computer output. 1. Verify that the numbers in c3 are a probability distribution.. Find the mean of the probability distribution in c3. 3. Find the standard deviation of the probability distribution in c3. You will now simulate your distribution 1 times. calc>randomdata>discrete In dialog box Generate 1 rows of data Store in columns c5-c14 Values in c1 Probabilities in c3 OK enter the commands rmeans c5-c14 c15 sort c15 c16 print c16 Make a stem and leaf display for c5 using the pull down menu or the command stem-and-leaf c5 describe c5 Make a stem and leaf display for c16 using the pull down menu or the command stem-and-leaf c16 describe c16 Make normal probability plots for c5 and c16. Answer the following questions on the sheet stapled to the front of your computer output. 1. Based on the stem and leaf display and the normal probability plot for c5 does the data appear to be normally distributed? Explain your answer.. Answer the same question for c What is the mean and the standard deviation obtained in the describe command for c5 and c16? 4. What should the mean and standard deviation be in theory for c5? for c 16? 5. Compare the mean and standard Deviation in questions 3 and 4 by finding the percentage error? 7

8 result(question3) result(question4) %error = result(question4) 6. State the Central Limit Theorem carefully and explain how the results you obtain in c16 validate it for your problem. :Double Integrals Review Problems Evaluate the Following Double integrals 1 1. (x + y)dydx ans 3 1. (x + y)dxdy ans /3 1 y 3. x 3 dydx ans 16/3 4 x 4. e (x +y)/ dydx ans 4 Find the areas of the following regions bounded by the indicated curves by double integration. Draw the regions first. 5. y = 5 x, y = ans 5/3 6. x 3y =, x + y = 5, y =. ans 5 7. y = x, y = x, x = ans Evaluate f (x,y)da for the regions given by the inequalities in problems 8-1. R Draw the regions first. 8. f(x,y) = 4xy < x <y < 1. ans 1/ 9. f(x,y) = 4xy, x 1, y 1, x + y 1. ans 1 1. f(x,y) = e -(x+y) x+y ans Answers to Selected Even Problems Section.3 3a. 4 b. 1 c. 18 d.13 e a. 6 b.1, 4.67,.333,.667 Section.4 5 a..5 b..1 c.56,.44 d..5 e.533 f..444, a..67 b,.59, 6.86 Section.5 74:.349,.651,(1 p) n,1 (1 p) n a..343 b..973 c..189 d,.16 e..353 Section a. 1/15 b..4 c..6 d. not a pdf. 16.a. f()=.41,f(1)=.4116, f()=.646, f(3)=.756,f(4)=.81 c. X=1 d a.. b..33 c..78 d..53 8

9 Section a..6, c mean = 3.4 sd = a. p b. p(1 p) c. p 34 Better to offer 4 copies (mean for 3 copies = 1.98, 4 copies.33) 38. EX=.3, Var(X)=.81 E(1 5X)=88.5 Var(1 5X)=.5 Section a..84 b..174 c..37 d. 4 Section x 1 x A. f(x) =,x =,1,,K1 5 1 B. g(x) = 1 (.3) x (.7) 1 x,x =,1,,K1 x C.exact EX = 3 V(X) = approximate EX = 3 V(X) =.1 7. P(3 kids)=1/4 P(4 kids)=3/8 P(5 kids)=3/8 (Hint look at World Series example on Web) Section a.3 b..973 C a..176 b..875 c a. mean = sd = 1.47 b..143 c y e A..569 B. 1!(y 1)! Section b. 865,.865,.135 c..471 d. P(X x) = 1 e x /θ,x > 8. c..18 d.9 e..74 f..4 Section 4. C. e ! n 18 a. (B A)p + A b. b. (B+ A) / c. σ x = (B A) / 1 d. B n k A k, k = y < y y < 5 A. F(y) = 5-1+ y 5 y 5 y y > 1 5p p < 1 B. µ = 1 5 (1 p) p 1 Section a..977 b..5 c..914 d e..417 f..687 (may be slightly different from answers obtained with tables. Was done on TI83,) 48. a..787 b..8643, a..993 b c

10 Section a..38 b..38 c..313 d..653 e..653 f..561 Section a..63 b..47 c a..714,.55 b..16 c d..86 Section 5.1 4a. expectation =1.7 b. p() =.19,p(1) =.3,p() =.8,p(3) =.3 1 b..5 c..36 1a..5 c..3 a.(1 e λt ) 1 b. 1 (1 e λt ) k λ (1 k)t e k 14. c.(1 e θt ) 9 (1 e λt ) 4 e 5λt + e θt 9 (1 e λt ) 4 e 4λt 4 5 Section 5. a b a. cov(x,y) = b. ρ =.7 Section a b Section Continuous Distribution Problems on sheet 1. A. g(x) = (1+ x), x 1 3 h(y) = ( 1 + y), y 1 3 B..417 C..417 D. cov(x,y) =.617, ρ =.818. A. not independent B..64 D., a b. no Section a. 63 b c. expectation = variance = d 14, Section a b c