Math/Stats 425-007 - Introduction to Probability

Winter 2020

Course Info

Course information sheet

Time and Location: MW 4:00-5:30 in East Hall 1068

Instructor: Corey Everlove

Email: everlove@umich.edu

Office: East Hall 4860

Final Exam

Here is the final exam and solutions.

To study for the final exam, you should try lots of practice problems, including the suggested problems from throughout the semester (in the schedule of classes below), the suggested Self-Test Problems below, the first two exams, and the old exams. Here is my final exam and solutions from last semester - skip problem 1f.

The Self-Test Problems at the end of each chapter have full solutions in the back of the book. Here are some suggested Self-Test Problems:

Class Schedule

Date Topics covered Suggested Problems Homework Notes
Tues, Apr 21 more examples from 7.2, 7.3, and 7.5
Wed, Apr 15 7.5: computing probabilities and expectation by conditioning Ch 7 probs: 48, 49, 53, 54, 56, 58
Mon, Apr 13 more 7.2 (computing expectation by breaking a RV into a sum of simpler RVs),
computing variance by a similar method (see 7.3 as well)		 Ch 7 probs: 15, 23, 35, 41
Compute the variance in problem 9a.
Thurs, Apr 9 more CLT (8.3)
7.2: linearity of expectation		 Ch 7 probs: 8, 9, 12, 13 Homework 7 and solutions
Tues, Apr 7 the law of large numbers
8.3: Central Limit Theorem		 Ch 8 probs: 5, 6, 8, 14, 15 The only section in Chapter 8 you should read (for now) is 8.3 (the Central Limit Theorem).
Wed, Apr 1 6.3: sums of independent random variables Ch 6 probs: 30, 31, 34
more problems
Mon, Mar 30 second midterm exam
Wed, Mar 25 6.4, 6.5: conditional distributions Ch 6 Probs: 38, 40, 41, 42,
Mon, Mar 23 more 6.1, 6.2 Ch 6 Probs: 13, 14, 16, 17, 18 Homework 6 and solutions
Wed, Mar 18 more 6.1: jointly distributed RVs
6.2: independence of RVs		 Ch 6 Probs: 19, 20, 21, 23, 28a
  • We will continue with material from Chapter 6 next week.
Mon, Mar 16 Review of normal distribution (5.4)
6.1: jointly distributed RVs		 Ch 5 Probs: 20
Ch 6 Probs: 2, 6, 9, 10
  • Class is now fully online.
  • Lectures can be found in the Files tab on our Canvas page.
  • More info on the upcoming "midterm" will be provided soon once I've decided on the details.
  • Please use our new Piazza site to ask questions about the lectures, suggested problems, or anything else!
Mon, Mar 9 5.4: the normal distribution and the normal approximation to the binomial distribution Ch 5 Probs: 16, 17, 22, 23, 24
  • The second midterm exam will be Wednesday, March 25.
  • While working on the suggested problems, you should figure out how to compute values of the normal CDF Φ(x) with the calculator you plan to use on exams.
Wed, Feb 26 More 5.1-5.3 (continuous RVs) Ch 5 Probs: 37, 38, 40, 41 Homework 5 and solutions
  • We talked about the St. Petersburg paradox at the start of class today.
  • Solutions to Homework 4 are now posted.
Mon, Feb 24 More on Poisson RVs (4.7),
5.1-5.3: basics of continuous RVs		 Ch 5 Probs: 1, 2, 4, 6, 10
Wed, Feb 19 4.7, 4.8: Geometric and negative binomial distributions, the Poisson distribution Ch 4 Probs: 55, 57, 59, 60, 71, 74 Homework 4 and solutions
  • The only distributions from this Chapter that you should know by name are the Bernoulli, binomial, and Poisson distributions. The others are important (geometric, negative binomial, etc.), but you do not have to remember their names.
  • The last sentence of suggested problem 74 should read "For part (b), what is the probability that the interviewer will have to talk to exactly (c) 6 people and (d) 7 people on the list in order to find 5 people who agree to be interviewed?"
Mon, Feb 17 4.4-4.6: Expectation and variance of RVs, Bernoulli and binomial RVs Ch 4 Probs: 20, 25, 28, 37, 42
Wed, Feb 12 (First midterm exam)
Mon, Feb 10 4.1-4.3: basic vocabulary of random variables and expectation (keep studying for exam)
Wed, Feb 5 3.4: independence of events, more problems The midterm exam I gave last semester and solutions.
  • A reminder that our first midterm exam will be next Wednesday, February 12 in class.
Mon, Feb 3 lots of problems involving conditional probability Start looking at the Self-Test Problems from Ch. 1-3.
  • We discussed the Monty Hall problem and a few variants.
  • I will have extra office hours Thursday and Friday 3-5.
  • You can turn in Homework 3 in class Wednesday, in office hours, or by email by 5pm on Friday.
Wed, Jan 29 3.3: conditioning, Bayes's formula Ch 3 Probs: 12, 24, 35, 38, 49 Homework 3 and solutions
Mon, Jan 27 3.2: conditional probability Ch 3 Probs: 1, 5, 6, 9, 10, 11
  • The first midterm exam will be Wednesday, February 12 (in class).
  • If you are finished with the homework, be sure to try some of the suggested problems (or any other problems that look interesting)!
Wed, Jan 22 more examples of computing probabilities by counting (Sec 2.5) Ch 2 Probs: 10, 28, 30, 37, 44 Homework 2 and solutions
  • We also discussed the inclusion-exclusion principle with 2 subsets/events as in Proposition 4.3, but not the general case (Proposition 4.4).
Wed, Jan 15 2.1-2.5: basic vocabulary of probability Ch 2 Probs: 5, 8, 15, 16, 23, 35
  • We didn't talk about the inclusion-exclusion principle (Props 4.3 and 4.4 in Section 2.4) yet.
  • Section 2.5 is the most important section in this chapter - be sure to read through many of the worked examples in this section.
  • I will have my usual 12-2 office hours on Monday even though there are no classes.
Mon, Jan 13 1.4-1.6: more combinatorics Ch 1 Probs: 27, 28, 30, 31, 34
Ch 1 Theoretical Exercises: 10		 Homework 1 and solutions
  • There is no class next Monday.
Wed, Jan 8 1.1-1.4: combinatorics Ch 1 Probs: 1, 3, 10, 16, 18, 19
  • Note that each chapter has Problems, Theoretical Exercises, and Self-Test Problems. The suggeted problems this time are all from the Problems section.
  • No homework yet!