Final Exam. Take-Home Portion. enter class. No answers will be accepted for any reason after 8:30. NO EXCEP-

Transcription

1 MATH 477 { Section E1 Final Exam Take-Home Portion The Rules: 1. The answers to this part of the exam will be due on Wednesday, July 29 th when you enter class. No answers will be accepted for any reason after 8:30. NO EXCEP- TIONS. You may submit answers sooner, of course. The preferred methods are to give it to a mailbox attendant in 315 Hill Center to put it in my mailbox ortoput it in a Manila envelope and slide it under my oce (606 Hill Center) door. Be sure either to call me ( ) or me (rymartin@math.rutgers.edu) so that I can let you know if I received it. If you submit the answers by class on Tuesday, I will return the graded answers to you at the end of class on Wednesday. 2. Write all answers neatly { if I can't read it, it's wrong. A good method is to write the answer to each problem on a dierent page. This way, you can work on the answers in any order you wish and then staple the nal result together. 3. Staple the grade template (the last page) to the front of your answers. 4. You may only consult the following materials: Your textbook, your class notes, previous homework that you have handed in, and any handouts that I have made and distributed. You may also use a calculus textbook, should you need a reminder. Forbidden materials include, but are not limited to: other classmates, other students (graduate or undergraduate), professors, any computing tool such as Maple, the Internet, other textbooks, journals, professors, the guy on the street that has a sign that says \Will do math for food." and so on. 5. I will continue to answer questions on homework, class material, etc.; but I will not answer questions, direct or indirect, that give solutions to this part of the exam. I will, however, clarify the statement of any question. 6. Solutions will be distributed on Wednesday, July 29, after class. Remember that the in-class portion will be during the regular class period (8:15-10:00) in the regular classroom (HH A1) on Thursday, July 30. Good luck! Make grading easy { submit a perfect paper. 1

2 Final Exam { Take-Home Portion (10 pts.) 1. A game show host has a contestant on stage and oers her three doors to choose. Behind one of those doors is a fabulous prize. Behind each of the other two isa copy of A First Course in Probability, 4 th Edition, by Sheldon Ross. The contestant rst chooses a door. The host will then open a door that she did not choose but that has a book behind it. (There is always at least one such door.) The host then says, \Would you like to stick with the door you chose originally, orwould you like to switch to the other one?" For example, she picks door #2 and the host says, \Well, if we open door #1, we see that it has no fabulous prize behind it. (But isn't that a lovely blue cover.) Would you like to stick with door #2 or would you like to switch to door #3?" Should she switch? Explain your answer. Assume that the fabulous prize is something much more desirable than the probability book. 2. Consider the following joint cumulative distribution function. F (x; y) =xy (1 + (1, x)(1, y)) 0 x 1; 0 y 1 : Compute the following: (3 pts.) a.) f(x; y), the joint p.d.f., (6 pts.) b.) f X (x) and f Y (y), the marginal p.d.f.s of X and Y, respectively, (6 pts.) c.) f XjY (x j y) and f Y jx (y j x), the conditional p.d.f.s, (4 pts.) d.) Cov(X; Y ), the covariance of X and Y, (8 pts.) e.) Var(X) and Var(Y ), (3 pts.) f.) (X; Y ), the correlation between X and Y. 3. Consider the following probability mass function: IP fx = i; Y = jg = p(i; j) =c 1, 2,i j i =0; 1;:::;N, 1 j =0; 1; 2;::: : Compute the following: (5 pts.) a.) c, (5 pts.) b.) p X (i), the marginal probability mass function of X, (5 pts.) c.) p Y jx (j j i), the conditional probability mass function of Y with respect to X = i, and 2

3 Final Exam { Take-Home Portion (5 pts.) d.) p Y (1), the marginal probability mass function of Y,evaluated at j =1. 4. Consider a sequence of n bits { numbers that are either 0 or 1. These represent a binary number as follows: If the sequence is (a 1 ;a 2 ;:::;a n ), then the number is For example: (0; 1; 1; 0; 1; 0; 0; 0; 1; 0) gives a a a n 2 n,1 : = 278 : We will create such anumber at random by making each bit 1 with probability p and 0 with probability 1, p. (10 pts.) a.) What is the expected value of the number that we create in this fashion? (10 pts.) b.) If all the bits are determined independently, what is the variance of the number that we create in this fashion? (10 pts.) 5. Two independent measurements, X and Y, are taken of a quantity. IE[X] = IE[Y ]=, but 2 6= X 2, where Y 2 =Var(X) and X 2 Y =Var(Y ). The two measurements are combined by means of a weighted average to give Z = X +(1, )Y where is a scalar and 0 1. Show that IE[Z] = and nd in terms of 2 X and 2 Y to minimize Var(Z). (10 pts.) 6. Prove that IE[X] =IE[X j X<a]IP fx <ag + IE[X j X a]ip fx ag : Hint: Dene an appropriate random variable and then compute IE[X] by conditioning on it. (10 pts.) 7. Let X have moment generating function M(t), and dene (t) = log M(t). Show that 00 (t)j t=0 =Var(X) : 3

4 Final Exam { Take-Home Portion (10 pts.) 8. For a standard normal random variable Z let n = IE[Z n ]. Show that n = 8 >< >: 0; when n is odd; (2j)! ; 2 j j! when n =2j. Hint: Start by expanding the moment generating function of Z into a Taylor series about 0. (10 pts.) Bonus. A die is continually rolled until the total sum of all rolls exceeds 600. Approximate the probability that at least 160 rolls are necessary. Hint: Use the result of the Central Limit Theorem. 4

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6 Name: MATH 477 { Section E1 Final Exam Take-Home Portion Problem Points Points Number Available Earned 1.) 10 2a.) 3 2b.) 6 2c.) 6 2d.) 4 2e.) 8 2f.) 3 3a.) 5 3b.) 5 3c.) 5 3d.) 5 4a.) 10 4b.) 10 5.) 10 6.) 10 7.) 10 8.) 10 Bonus (10) Total 120 6