Math351 Spring 2010

Exams

mikhailmatveev — Fri, 12 Feb 2010 22:28:44 +0000

Exam 1 will be on Friday, February 19 and will cover Chapters 1 through 3. Exam1Solution

Exam 2 will be on Friday, March 19 and will cover material up to Section 5.2 (inclusively). The exam will consist of five problems (you will be supposed to solve all of them). Topics of the problems:

- Basic notions about discrete and continuous random variables : probability mass function (for discrete), density (for continuous), cumulative distribution function (for both), expectation, variance, standard deviation. Expectation of a sum of variables.

- Bernoulli, binomial, negative binomial, geometric, and hypergeometric variables. Formulas for their expected values.

- Poisson variable. You will need a calculator that can calculate exponents.

The exam will be conducted by another professor. I will be available to answer questions on Wednesday, but I will leave the office about 4pm. On Thursday or Friday I will not be available.

Exam2Solution

If you did not do well in Problem 2 of Exam 2 (and did well in other problems), then you may solve a make-up Problem 2, and increase your grade for Exam 2. If you want to try this, find me in the office any Monday or Wednesday some time after the class (let us finish with this until the end of April). Warnings: the make-up problem will be slightly more difficult than the one in the exam, and the maximum you can get for it will be 13 points (instead of 15 in the exam)

Exam 3 on Friday, April 16 will be based on Chapters 5 and 6 of the book. You will need a calculator that can calculate exponents, logarithms, square roots, etc. The exam paper will contain a copy of Table 5.1 from the book (area under standard normal curve)

Exam3Solution

Exam 4 on Monday, May 3 will be based on Chapter 7. The format of the exam will be similar to exams 2 and 3. You may need a calculator. Principle (new) skills:

- Expressing a random variable as a sum of several simpler variables (in order to find the expectation);

- Understanding covariance, variance of sums, etc.

- Understanding conditional expectation.

Moment Generating Functions (Section 7.7) will not be included in this exam. (But please feel free to use this technique for solving problems if you find it appropriate. All problems in the exam, however, will be solvable without Moment Generating Functions.) Moment Generating Functions may be included in the selective part of the Final Exam.

Exam4Solution

The final exam will be similar in format to Exam 1. It will consist of two parts. Part 1: relatively simple problems, you will be supposed to solve all of them. Part 2: more difficult (or more exotic) problems. You will be supposed to choose several problems (say 4 from 6; this will be specified) and solve them. No extra credit for solving more problems than required.

FinalReview (without one picture)

On Monday, May 10 and Tuesday, May 11, I will be in the office from about noon until about 8PM. Please come with questions if you like.

FinalReviewWithQuestionsAnswered

Classes (and homework for the next time)

mikhailmatveev — Wed, 20 Jan 2010 19:19:35 +0000

January 20 Book: sections 1.1-1.3 Homework from the book: Chapter 1: P1, P4, P7, P8(a,c), P11, T2, ST1, ST2

To someone who forgot the book in class: pick it in my office, or otherwise I will bring it on Friday.

Attention! Please note the correction of the final exam date and time in the Syllabus.

January 22. Book: section 1.4. Homework from the book: Chapter 1: P13, P15, P19(a), P21, P24, P25, T6, T8, T10, T11, T17, ST3, ST7

January 25. Book: section 1.5. Homework from the book: conclude the homework for Chapter 1 (see the list).

January 27. Recitation on Chapter 1, started Chapter 2 (sections 2.1, 2.2) Homework from the book: Chapter 2: P1, P2, P3, P4, P7, T6, T7. AnswersBookChapter1

January 29. Book: sections 2.3, 2.4. Homework from the book: P8, P9, P12, T13

February 01. Discussed 2.5. Homework: read 2. 5, read or skip 2.6, read 2.7. (Handout +) conclude the exercises for Chapter 2 of the book (see the list below). Next time we will have recitation on Chapter 2 and may be will begin Chapter 3.

February 03. Recitation on chapter 2, started Chapter 3. Homework: (handout +) read 3.1, 3.2 AnswersBookChapter2

February 05. Class canceled because the University is closed for weather problems.

February 08. Class canceled because the University is still closed for weather problems. Exam 1 (which I would like to be based on the first three chapters) is postponed… we will decide together for how long after the University re-opens.

Byong Kwon, a student in this class, sent me this: matt-parker-head-tails He assumed this might be interesting to other students. I also think it might. Have fun!

February 10. Still closed. The following problem might be a good extra exercise on the notion of conditional probability: RussianRoullette Have fun!

February 12. Book: section 3.2 (concluded), 3.3. Homework: handout+ problems from the list tforChapter 3 except those that involve independence. Exam 1 will be on Friday, February 19.

February 15. Considered independence (section 3.4 of the book). Homework: handout + read 3.4 and 3.5 (in 3.5, it is enough if that you read only the beginning of the section to understand that conditional probabilities satisfy the same rules as the “absolute” probabilities) + conclude the list of problems for Chapter 3. Attention! If you were not in class today, please check the updated syllabus (extended semester, times for exams, etc.)

February 17. Review of Chapter 3, and briefly discussed section 4.1. Homework for Feb. 22: Handout + read 4.1. AnswersBookChapter3

February 19. Exam 1, no new homework.

February 22. Discussed discrete random variables, probability mass function, cumulative distribution function, expectation. Homework: 4.2, 4.3, 4.10. Handout + problems from the book 4.1, 4.2, 4.4, 4.19, 4.22(a). 4.25, 4.27, 4.28

February 24. Discussed expectation (cont’d) and variance of random variables. Homework: sections 4.4, 4.5, handout, problems from the book 30, 35, T2, T3, T8.

February 26. Discussed Bernoulli and binomial random variables (Section 4.6) , and addition for random variables (section 4.9). Homework: 4.6, 4.9, Problems: handout + from the book P40, P56, ST9, ST21

March 01. Discussed Geometric, Negative binomial, and Hypergeometric random variables (Section 4.8 of the book; in this section, you may skip subsection 4.8.4. We do not skip section 4.7, we will discuss it next time.) Homework: handout + from the book: 28, 75, 79, 82, T27

March 03. Discussed Poisson distribution. Homework: read 4.7, Problems: handout + conclude problems from Chapter 4 of the book (see the list). Next time we will make a recitation on Chapter 4.

March 05. Recitation on Chapter 4. AnswersBookChapter4 Homework: only one turn-in problem (see below).

March 15. 5.1, 5.2 Homework: handout + P1, P6(b), T7, ST1

March 17. 5.3 Homework (for Monday) P11, 12, 13, T27

March 19 Exam 2

March 22. 5.4 (except Normal Approximation of Binomial Distribution). Homework: 5.15, 5.16, 5.18, 519, T9, St8

March 24. Discussion of the exam, finished section 5.4 (Normal Approximation of the Binomial Distribution) Homework: handout and 5.27

March 26. Discussed 5.5 (except Hazard Rate), 5.7. Homework: handout + 5.32, 5.33, 5.40, ST13, T29, and This problem: let X be exponential r.v. with parameter lambda. Find f_Y and F_Y where Y=ln(X). When solving problem (1) from the handout, be careful: unlike in the example that we considered in class, the function (X -> Y=|X|) is not one-to-one. It is a good idea, that before solving this problem, you look also at the examples in Section 5.7 in the book, not only the lecture notes…

March 29. Concluded Section 5.5 (discussed hazard rates). Next time we will begin Chapter 6 (there will be a recitation on Chapters 5 and 6 together before Exam 3). Homework: handout + 5.35, 5.36 (these two problems are about hazard rate; but perhaps now it makes sense to conclude all homework for Chapter 5 of the book)

March 31. 6.1 Homework: handout + 6.1(c), 6.2(a), 6.10(a), T6, ST3. AnswersBookChapter5

April 02. 6.2 Homework: handout + 6.13, 6.22, 6.26, T3, ST7. Please check the list of homework exercises and stuff to skip posted in “Problems from the Book” below

April 05. 6.3. Homework exercises: handout only.

April 07. 6.4. Homework: handout

April 09. 6.6 Homework: handout + problem 6.41 from the book. Plan for the next week: Monday, April 12: 6.7 (It will help if you review the topic “Change of variables in double integrals” if you still have the calculus book). We skip sections 6.6 and 6.8. Wednesday, April 14: recitation on Chapters 5 and 6. Friday, April 16: Exam 3.

April 12. (a review of 5.7+) 6.7. Homework: handout only. A misprint was noticed in the handed out homework assignment: in “10(less or equal) x_2 (less or equal) 10″ the first “10″ should be “0″. A corrected assignment is posted below in the homework section. AnswersBookChapter5 AnswersBookChapter6

April 14. Recitation on Chapters 5 and 6. Please be ready for the exam on Friday. I am posting the next homework assignment in the Homework section below, but it is due next Monday.

April 16. Exam 3.

April 19. 7.1 and a part of 7.2 (We considered expectation of functions of random variables in general. Next time we will consider the special case when the function is a sum of random variables) Homework: handout + problems 1 and 4 from Chapter 7 of the book. When reading Section 7.2, you may skip Examples 2i, 2k, 2m, and the end of the section starting from 2p.

April 21. Finished 7.2, 7.3. Homework: handout + problems P6, P7, P9, P12 from Chapter 7 of the book. When reading Section 7.3, you may skip Examples 3b, 3d, 3e, 3f.

April 23. 7.4 Homework: handout + problems P30, P33, P37, P38, T22 from Chapter 7 of the book. When reading Section 7.4, you may skip Example 4c.

April26. 7.5 Homework: Handout + P48(a,b), P50.

April 28. 7.7 Homework: Handout + P76, T48 AnswersBookChapter7

April 30. Discussed Chapter 8. Will finish with it on Wednesday. Please be ready for the exam on Monday! Homework (for Wednesday: Handout)

May 03. Exam 4. No new homework. The two remaining classes will be (except for 15 min to finish with Chapter 8 on Wednesday) review sessions. I will have some list of points to discuss, but please prepare your questions too.

May 05. Exam 4 discussion, finished Chapter 8, Final review handed out. (See the file in “Exams”)

May 07. Final review