Umich Math 425 Homework 5 Solutions
Math 425: Introduction to Probability
Fall 2010, Section 3
Course meets: Tuesday and Thursday 1:102:30 in 1084 East Hall.
Instructor: Sergey Fomin, 2858 East Hall, 7646297, fomin@umich.edu
Office hours: Tuesday 3:004:00 and Thursday 4:105:30 in 2858 East Hall. No office hours on 9/23, 10/14, 11/4, 12/9.
Grader: Yingying Tan, yings@umich.edu
Course homepage: http://www.math.lsa.umich.edu/~fomin/425f10.html
Text (required): Sheldon Ross, A First Course in Probability, 8th edition, PrenticeHall, 2010.
Where can I buy this textbook? (also check out publisher's online store and international editions)
Prerequisites: Math 215 or 285 (Multivariable calculus).
Course synopsis:
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 17 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).
Homework
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 (1^{st} part), 51; Chapter 8, 7, 13(a), 15.
Exams
Exams are closed book, closed notebook. You will be allowed to bring a 3by5 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 13.
Practice problems for Exam #1 and answers to them. Another practice problem
Exam #2 covers Chapters 45, with the exception of Sections 4.8.34.8.4, 5.5.1, 5.6.
Practice problems for Exam #2: SelfTest 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 6.16.5, 6.7, 7.17.2 (except 7.2.17.2.2), 7.47.5 (except 7.5.27.5.4), 8.18.3, and 9.1.
Practice problems for the final exam and answers to them.
Exam times
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, 46 PM, in 1084 East Hall.
(Time of the final exam is determined by the Office of the Registrar.)
Time & Location  Professor  Office hours 

[ Announcements ] [ Course Description ] [ Schedule ]
Announcements
Course Description
Prerequisites
Three term calculus sequence. The course will make essential use of the material of Math 116 and 215.
Text
A first course in probability by Sheldon Ross (Seventh edition).Content
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 18 from the textbook.
Grading
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.
Homework  25% 
Midterm 1  20% 
Midterm 2  25% 
Final  30% 
Schedule
Lecture  Day  Topic  Section  Notes  Homework 

1  Jan 7  Basic Principles of Counting  1.13  Notes  
2  Jan 9  Binomial Coefficients  1.45  Notes  Homework 1 (due Jan 21) 
3  Jan 12  Review of Chapter 1  Notes  
4  Jan 14  Probability Models  2.12  Notes  
5  Jan 16  Axioms of Probabilities  2.34  Notes  
Jan 19  NO CLASS  
6  Jan 21  Sample spaces with equiprobable outcomes  2.5  Notes Birthday applet  Homework 2 (due Jan 28) 
7  Jan 23  Inclusion/Exclusion Principle Review of Chapter 2  Notes  
8  Jan 26  Conditional Probability  3.12  Monty Hall Paradox Notes  
9  Jan 28  Bayes Theorem  3.3  Notes  Homework 3 (due Feb 4) 
10  Jan 30  Independence  3.4  Notes  
11  Feb 2  Conditional probability function  3.5  Notes  
12  Feb 4  Review of Chapter 3  Notes  
13  Feb 6  Discrete Random Variables  4.12  Notes Benford's Law  
14  Feb 9  Expectation  4.34  Notes Sic Bo  
15  Feb 11  Midterm 1  Chapters 1  3  Midterm 1 guide  Homework 4 (due Feb 18) 
16  Feb 13  Variance  4.45  Notes  
17  Feb 16  Bernoulli distributions  4.6  Binomial Calculator Notes  
18  Feb 18  Poisson distributions  4.7  Poisson Calculator Notes  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  5.12, 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 45  Midterm 2 guide  Homework 8 (due April 1) 
31  March 27  Conditional distributions  6.45  Notes  
32  March 30  Joint probability distributions  6.7  Notes  
33  April 1  Review Chapter 6  Notes  Homework 9 (due April 8)  
34  April 3  Expectation  7.12  Notes  
35  April 6  Expectation  7.2  Notes  
36  April 8  Covariance  7.4  Notes  Homework 10 (due April 15) 
37  April 10  Conditional expectation  7.56  Notes  
38  April 13  Chebyshev's inequality and the Weak Law  8.12  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 46pm 
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