MATH 323 Fall 2020 Final Exam Solutions PDF

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Faculty of ScienceWMtrffia&&Final ExaminationFall 2020 lntroduction The purpose of this exam is to provide you with an opportunity to demonstrate what you have learned in this course. Because of the challenging situation imposed by COVID-19, the Faculty of Science wants to ensure that you ar...

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Instructions 1. Either use the template to answer your questions or answer on a separate sheet of paper. If you answer on a separate sheet of paper, your answers must be ordered as follows: – Question 1 – Question 2 – Question 3 – Question 4 – Question 5 – Question 6 Your answers must be listed in this order, even though you may have worked on them in a different order. Thus, for example if your first page contains the answer to Question 2 instead of Question 1 you will receive no marks for your answers. 2. Your exam must be submitted by any means to myCourses as a single pdf (handwritten or typed) , no later than December 11, 2020 at 13:59 (1:59 pm). 3. Submit your exam on myCourses as if you are submitting an assignment.Make sure that you submit your exam to “Final Exam”and not any other folder. 4. You will be allowed multiple submissions until the deadline. We will grade the latest submission before the deadline. 5. Make sure that you receive an acknowledgement from myCourses that your submission has been received. If you encounter a problem with your submission you must let one of the course instructors know immediately. It is too late to inform one of the instructors once the deadline has passed. 6. It is recommended that you do not take the full 72 hours to complete your exam. Do not leave your submission until shortly before the deadline 7. You must briefly show your reasoning.

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(15) 1.

a. (3 Points) Assume that the probability of a new born child being a boy is 1 − p and the probability of being a girl is p. Assume that the sex of a newborn child is independent of the sex of previous children (if any) with the same parents. A married couple decides to have children until a girl is born. What is the probability mass function of X, the number of children needed to have a girl? b. (1 Point) Again, if the probability of a girl is p, what is the expected value of X? c. (7 Points) Another couple, with the same probabilities of having a boy or girl, decides to continue having children until they have at least one child of each sex. What is the distribution of the number of children, Y , the couple must have for this to occur? d. (4 Points) What is the expected value of Y ?

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(25) 2.

a. (10 Points) A box contains b black marbles and r red marbles. One of the marbles is drawn at random, but when it is put back in the box, c additional marbles of the 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 b drawn is red, is b+r+c . b. (5 Points) Suppose there are two tests for Covid-19 performed by two different 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 What is the probability that the swab will be declared positive by both companies? The information that has been collected on these two tests gives you: 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 down the probability that at least 3 will be kept for strictly longer than three years. Your answer should be written in terms of F .

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(15) 3.

a. (3 Points) A disease affects 6.5% of the population. There is however, an inheritance 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 different 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 different ages who have written in to the show. If the participants are selected randomly,without replacement, what is the probability that exactly two of the five youngest people on the list will be selected ?

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(30) 4. Let the random variables Y1 and Y2 have joint probability density function: ⇢ 2 cy1 y2 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1 f (y1 , y2 ) = 0 elsewhere Find: a. (2 Points). The constant c b. (4 Points) The marginal density function for Y1 and Y2 c. (5 Points) P (Y1 ≤ 1/2|Y2 ≥ 3/4) d. (5 Points) The conditional density function of Y1 given Y2 = y2 e. (6 Points) P (Y1 ≤ 3/4|Y2 = 1/2) f. (4 Points) E[Y1 ], E[Y2 ], V ar [Y1 ], andV ar [Y2 ] g. (5 Points) Corr(Y1 , Y2 )

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(20) 5.

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 randomly 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 variance is 100. Find the approximate probability that the average monthly number of new cases of MS over a 10 year period is at least 11.5. Show your reasoning.

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(30) 6.

a (5 Points) Is the following f(x) a probability density function? If so, evaluate the cumulative distribution function. 8 0≤x≤1...