math-323-fall-2020-final-exam-solutions

Faculty of Science weffitr#t}a 4. Exam time period and submissioni lnstructor please read and complete. a) The exam ha$-been prepared in such a waV that..L{__..l1..,! ertaken under normal conditions, it can be completed in #hours. You have been allotteO iffif'ours to access the exam and , 7i hours to complete and submit the exam with your answers once you begin. Due to potential issues with internet connectivity and other unforeseen technical issues, it is strongly recommended that you plan to submit your exam script well in advance of the deadline. See also c) below. b) lf the access or completion deadline falls on a weekend day, the deadline remains. c) For some exams, you may be allowed to submit an exam script more than once during the allotted time; only your last submission will be graded (regardless of its state of completion). d) All exams must be submitted by the posted deadline. e) Please check your exam script for readability and completeness before uploading and submitting. 5. Questions that arise during the exam The type of questions to which an instructor can respond during an exam may be limited lnstructor: Pleose describe what kinds of questions you will onswer. See example given The Examiner will be responding to questions: 5. Academic integrity 0,* S4tr l McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures ( see www.mcgiil.ca/students/srr/honest/ for more information, a pproved by Senate on January 1.6,2019) Please affirm the following statement by either uploading a signed copy of this form; digitally signing a statement at the beginning of your submission; sending an email within myCourses; taking a photo of your signature and uploading it with the exam script; (if appropriate) responding to a 1-question quiz within myCourses that acknowledges acceptance of the integritycode. I hove neither given nor received unouthorized oid on this exam and I agree to adhere to the specific Terms and Conditions thot govern this exom. Student: please sign here Signature: lf you find.an exam question to be ambiguouS'or unclear, email your concerns to the instructor at the address shown above in the General lnformation section. Your instructor will try to review questions periodically during the exam period, but a response cannot be guaranteed. lt is important, therefore, that you express any concerns over ambiguities on your exam script and still answer questions to the best of your ability. E rt i i, ::: Prof. Wolfson: L0am to 11am on Decemberg, 10 and 11. Dr. Sajjad: 2pm to 3pm on December 9 and 10 and 11am to Downloaded by Nicholas Sun (sun_nicholas@yahoo.com) lOMoARcPSD|16672699

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2. (25) a. (10 Points) A box contains b black marbles and r red marbles. One of the marbles isdrawnatrandom, butwhenitisputbackinthebox, c additionalmarblesofthe same colour are put in with it. Now suppose that another marble is drawn. Prove that the probability that the first marble drawn was black if the second marble drawn is red, is b b + r + c . b. (5 Points) Suppose there are two tests for Covid-19 performed by two di ff erent companies, A and B. Nose swabs taken from people are sent first to company A and then company B . Company A does not reveal the outcome of it’s tests to company B . Suppose that the tests are based on similar biological mechanisms. Let A + = Event that A declares a swab positive and B + = Event that B declares a swab positive Whatistheprobabilitythattheswabwillbedeclaredpositivebybothcompanies? Theinformationthathasbeencollectedonthesetwotestsgivesyou: P ( A + )=0 . 1, P ( B + )=0 . 08. Given the above information, if possible, find the probability that a swab will test positive at both companies, showing your reasoning. If you think it is not possible to find this probability, say why. c. (10 Points) Let X be the time (in years) that a certain laptop computer is kept before being replaced. Let F ( x ) be the cumulative distribution function of X . If a sample of 10 laptops is selected at random from millions of such laptops, write downtheprobabilitythatatleast3willbekeptforstrictlylongerthanthreeyears. Your answer should be written in terms of F . Page 8 Downloaded by Nicholas Sun (sun_nicholas@yahoo.com) lOMoARcPSD|16672699

3. (15) a. (3 Points) A disease a ff ects 6 . 5% of the population. There is however, an inheri- tance factor. If one’s father has the disease, the probability that the child will get the disease is 0 . 13. What is the probability a child will get the disease when the father does not have the disease? b. (6 Points) A car dealer has 9 cars on the lot. Each car is a di ff erent make. She selects three cars at random with replacement. What is the probability that at least two of the cars in her sample will be the same make? c. (6 Points) The organizer of a television show must select 5 people to participate in the show. The participants will be selected from a list of 28 people with dif- ferent ages who have written in to the show. If the participants are selected ran- domly,without replacement, what is the probability that exactly two of the five youngest people on the list will be selected ? Page 11 Downloaded by Nicholas Sun (sun_nicholas@yahoo.com) lOMoARcPSD|16672699

4. (30) Let the random variables Y 1 and Y 2 have joint probability density function: f ( y 1 ,y 2 )= ⇢ cy 2 1 y 2 0 ≤ y 1 ≤ 1 , 0 ≤ y 2 ≤ 1 0 elsewhere Find: a. (2 Points). The constant c b. (4 Points) The marginal density function for Y 1 and Y 2 c. (5 Points) P ( Y 1 ≤ 1 / 2 | Y 2 ≥ 3 / 4) d. (5 Points) The conditional density function of Y 1 given Y 2 = y 2 e. (6 Points) P ( Y 1 ≤ 3 / 4 | Y 2 =1 / 2) f. (4 Points) E [ Y 1 ] ,E [ Y 2 ] ,Var [ Y 1 ] ,andVar [ Y 2 ] g. (5 Points) Corr ( Y 1 ,Y 2 ) Page 14 Downloaded by Nicholas Sun (sun_nicholas@yahoo.com) lOMoARcPSD|16672699

5. (20) a. (3 Points) An automatic device fills capsules that are each supposed to contain 10 mg of medication. It is found that the amount that actually goes into each capsule is a Normal random variable whose mean is 10 mg and whose standard deviation is 0.5 mg. One such capsule is selected at random. What is the probability that it contains at least 10.1 mg of medication? b. (7 Points) What is the probability that the total weight of medication in 10 ran- domly selected capsules will be at least 90 mg? An exact numerical answer is required. Show your reasoning. c. (10 Points) Suppose that the number of new cases per month of multiple sclerosis (MS) in a very large population, is a random variable whose mean is 10 and whose varianceis100. Findtheapproximateprobabilitythattheaveragemonthlynumber of new cases of MS over a 10 year period is at least 11.5. Show your reasoning. Page 17 Downloaded by Nicholas Sun (sun_nicholas@yahoo.com) lOMoARcPSD|16672699

6. (30) a (5 Points) Is the following f(x) a probability density function? If so, evaluate the cumulative distribution function. f ( x )= 8 < : x 0 ≤ x ≤ 1 2 - x 1 ≤ x ≤ 2 0 elsewhere b. Itisknownthatthetimesinhoursbetweentra ffi caccidentsatacertaindangerous intersectionareindependentandfollowaGammadistributionwhosemeanis8and whose variance is 4. i. (4 Points) Write down (but do not derive) the moment generating function of the time between tra ffi c accidents at this intersection. ii. (5 Points) Suppose a tra ffi c accident occurs at noon. Prove that the moment generatingfunctionofthetime, Y ,fromnoon,tothe5thtra ffi caccident after noon is given by M Y ( t )= ⇣ 2 - t 2 ⌘ - 16 for t< 2. iii. (1Point). Whatisthenameofthedistributionimpliedbyyouranswertopart (ii)? iv. (5Points)Usingthemomentgeneratingfunctionfrompart(ii)derivethemean and variance of the time from noon to the 5th tra ffi c accident after noon. v. (10 Points) Suppose now, that you are told that the time between tra ffi c ac- cidents has a Gamma distribution with α = 1 and β = 4. You arrive at the intersection which is accident free at that time. What is the probability that the next accident after your arrival will occur within 3 hours? A numerical answer is required. Page 20 Downloaded by Nicholas Sun (sun_nicholas@yahoo.com) lOMoARcPSD|16672699

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Solution: Page 22 Downloaded by Nicholas Sun (sun_nicholas@yahoo.com) lOMoARcPSD|16672699