Umich Math 425 Homework 5 Solutions
Math 425: Introduction to Probability
Fall 2010, Section 3
Course meets: Tuesday and Thursday 1:10-2:30 in 1084 East Hall.
Instructor: Sergey Fomin, 2858 East Hall, 764-6297, email@example.com
Office hours: Tuesday 3:00-4:00 and Thursday 4:10-5:30 in 2858 East Hall. No office hours on 9/23, 10/14, 11/4, 12/9.
Grader: Yingying Tan, firstname.lastname@example.org
Course homepage: http://www.math.lsa.umich.edu/~fomin/425f10.html
Text (required): Sheldon Ross, A First Course in Probability, 8th edition, Prentice-Hall, 2010.
Where can I buy this textbook? (also check out publisher's online store and international editions)
Prerequisites: Math 215 or 285 (Multi-variable calculus).
This course introduces students to the theory of probability and to a number of applications. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. The material corresponds to most of Chapters 1-7 and part of 8 of Ross.
Grade will be based on two midterm exams (held during regular class time), 25% each; 20% homework; 30% final exam (comprehensive).
Your lowest homework set score will be dropped.
This course will not be graded on a curve, i.e., there are not a set number of each grade to be given out. Every student with the total score of 90% (resp., 80%, 70%, 60%) is guaranteed the final grade of A (resp., B or higher, C or higher, D or higher).
There will be 10 problem sets. No late homework will be accepted. In each homework assignment, 5 problems will be graded. All answers should be justified by a sound argument. An answer lacking justification will receive no credit. Collaboration on the homework is fine, but each person is responsible for writing up her/his own solutions. Due dates shown below are tentative; actual due dates will be announced in class.
HW#1, due 9/14: Chapter 1, problems 3, 8(c), 12(a), 15, 22, 24, 26, 28, 30.
HW#2, due 9/21: Chapter 2, 12(a), 17, 21(b), 29(a), 32, 42, 43(b), 45, 47.
HW#3, due 9/28: Chapter 3, 4, 13, 16, 18(a), 26, 36, 47, 53.
HW#4, due 10/05: Chapter 3, 39, 48, 56, 62, 63(c), 66, 70, 71.
HW#5, due 10/21: Chapter 4, 4, 14, 22(b), 32, 38, 41, 43, 48.
HW#6, due 10/28: Chapter 4, 54, 64(b), 65(b); Chapter 5, 1, 5, 7, 11, 13(b).
HW#7, due 11/4: Chapter 5, 18, 19, 21, 24, 27, 32, 33.
HW#8, due 11/23: Chapter 6, 6, 8, 14, 27, 29(a), 30(a), 31(a).
HW#9, due 12/2: Chapter 6, 41(a), 58; Chapter 7, 9(a), 12, 19(a), 34(a).
HW#10, due 12/9: Chapter 7, 34(b), 39, 42 (1st part), 51; Chapter 8, 7, 13(a), 15.
Exams are closed book, closed notebook. You will be allowed to bring a 3-by-5 index card to the 1st midterm, two such cards to the 2nd midterm, and three cards to the final. One problem on each midterm exam will be taken directly from homework (perhaps with altered numerical values).
Past exams can be found on Math 425 pages by Dan Burns and Hugh Montgomery.
The midterm exams are held in class. No makeups will be given.
Exam #1 covers Chapters 1-3.
Practice problems for Exam #1 and answers to them. Another practice problem
Exam #2 covers Chapters 4-5, with the exception of Sections 4.8.3-4.8.4, 5.5.1, 5.6.
Practice problems for Exam #2: Self-Test Problems and Exercises in Chapter 4, 3, 13, 15; and in Chapter 5, 11, 12
Final exam covers the topics covered by the midterm exams (see above), plus Sections 6.1-6.5, 6.7, 7.1-7.2 (except 7.2.1-7.2.2), 7.4-7.5 (except 7.5.2-7.5.4), 8.1-8.3, and 9.1.
Practice problems for the final exam and answers to them.
All exams will be held in the same room where the class meets.
Midterm exam dates shown below are tentative; actual dates will be announced in class.
First midterm: October 7.
Second midterm: November 9.
Final exam: December 15, 4-6 PM, in 1084 East Hall.
(Time of the final exam is determined by the Office of the Registrar.)
Time & Location
[ Announcements ] [ Course Description ] [ Schedule ]
Three term calculus sequence. The course will make essential use of the material of Math 116 and 215.
TextA first course in probability by Sheldon Ross (Seventh edition).
This course introduces students to useful and interesting ideas of the mathematical theory of probability. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. We will cover most of chapters 1-8 from the textbook.
There will be 2 midterms (in class), and a final. We will have 11 weekly homework assignments. Grades will be determined by the following weighting.
|1||Jan 7||Basic Principles of Counting||1.1-3||Notes|
|2||Jan 9||Binomial Coefficients||1.4-5||Notes||Homework 1 (due Jan 21)|
|3||Jan 12||Review of Chapter 1||Notes|
|4||Jan 14||Probability Models||2.1-2||Notes|
|5||Jan 16||Axioms of Probabilities||2.3-4||Notes|
|Jan 19||NO CLASS|
|6||Jan 21||Sample spaces with equiprobable outcomes||2.5||Notes|
|Homework 2 (due Jan 28)|
|7||Jan 23||Inclusion/Exclusion Principle |
Review of Chapter 2
|8||Jan 26||Conditional Probability||3.1-2|
Monty Hall Paradox
|9||Jan 28||Bayes Theorem||3.3||Notes||Homework 3 (due Feb 4)|
|11||Feb 2||Conditional probability function||3.5||Notes|
|12||Feb 4||Review of Chapter 3||Notes|
|13||Feb 6||Discrete Random Variables||4.1-2||Notes|
|15||Feb 11||Midterm 1||Chapters 1 - 3||Midterm 1 guide||Homework 4 (due Feb 18)|
|17||Feb 16||Bernoulli distributions||4.6|| Binomial Calculator|
|18||Feb 18||Poisson distributions||4.7|| Poisson Calculator|
|Homework 5 (due March 4)|
|19||Feb 20||Other distributions||4.8||Notes|
|Spring Break, No class on week of Feb 23 - 27|
|20||March 2||Continuous random variables||5.1-2, 5.7||Notes|
|21||March 4||Uniform distributions||5.3||Notes||Homework 6 (due March 11)|
|22||March 6||Normal distributions||5.4||Notes|
|23||March 9||Normal distribution||5.4||Notes|
|24||March 11||Exponential distribution||5.5||Notes||Homework 7 (due March 18)|
|25||March 13||Other continuous distributions|
|26||March 16||Joint distributions||6.1|
|28||March 18||Independent R.V.s||6.2||Notes|
|28||March 20||Sums of R.V.s||6.3||Notes|
|29||March 23||Sums of R.V.s||6.3|
|30||March 25||Midterm 2||Chapters 4-5||Midterm 2 guide||Homework 8 (due April 1)|
|31||March 27||Conditional distributions||6.4-5||Notes|
|32||March 30||Joint probability distributions||6.7||Notes|
|33||April 1||Review Chapter 6||Notes||Homework 9 (due April 8)|
|36||April 8||Covariance||7.4||Notes||Homework 10 (due April 15)|
|37||April 10||Conditional expectation||7.5-6||Notes|
|38||April 13||Chebyshev's inequality and the Weak Law||8.1-2||Notes|
|39||April 15||Central Limit Theorem||8.3||Notes||Homework 11 (due April 27)|
|40||April 17||Strong Law of Large Numbers||8.4||Notes|
|41||April 20||Chapter 8 Review|
|April 22||Exam Review|
(Not a required class day)
|April 27||Final Exam|
Monday, April 27