MATH 461, Sections B/C, Spring 2009

Final Exam and Course Grades

The letter grade shown at the end of the score display is your course grade. These grades were assigned on a linear scale, based on the accumulated total score. To avoid hardships and "close calls", I adjusted some of cutoffs. As a result, nobody was within a few points of the next higher grade, and most gaps between grades were in the order of 15 points or more.

Final Exam solutions are available under the following links:

• Section B (9 am) Final Exam Solutions.
• Section C (10 am) Final Exam Solutions.

Have a good break and enjoy your summer!

Course Policies, Exams, Grades

• Course Information Sheet. The first-day handout. Everything you need to know about this class: Syllabus, grading policies, office hours, etc.
• Online Scores. Log in with your Net-ID and password. The display shows the scores on all assignments and exams given out so far. Any score discrepancies must be reported within one week of the assignment/test.
  [5/2/09] Exam 3 scores are in. See the Exam 3 Results and Grading page for an explanation of the score display, grading policies, etc.
• Final Exam:
  • Final Exam Information. Everything you need to know about the Final. Syllabus, format, grading information, etc.
  • Dates, times, and locations:
    • Section B (9 - 9:50 am): Thursday, May 14, 8 am - 11 am, 347 Altgeld.
    • Section C (10 - 10:50 am): Monday, May 11, 8 am - 11 am, 1310 DCL.
• Midterm Exam 3: Thursday, 4/30, 7 pm - 8 pm, Room 1404 Siebel
  • Exam 3 Information and Syllabus.
  • Exam 3 Solutions.
  • Exam 3 Results and Grading Information.
• Midterm Exam 2: Thursday, 3/19, 7 pm - 8 pm, Room 1404 Siebel.
  • Exam 2 Information and Syllabus.
  • Exam 2 Solutions.
  • Exam 2 Statistics and Grading Information. Check here for exam statistics (average scores, etc.) and grading information.
• Midterm Exam 1: Thursday, 2/19, 7 pm - 8 pm, Room 1404 Siebel.
  • Exam 1 Information and Syllabus.
  • Exam 1 Solutions.
  • Exam 1 Statistics and Grading Information.

Announcements

• [1/26/09] Open House to begin this week: I will begin to hold weekly Open House hours, Wednesdays, 5 pm - 6 pm (longer if needed), 141 Altgeld. The Open House is intended as an informal office hour for students in this class; take advantage of this opportunity!
• [1/26/09] Exam schedule: Midterm exams are scheduled as follows. Note that the exams are in a different, much larger, and much nicer, room in Siebel Center, the brandnew and ultramodern CS building, located between Goodwin Ave. and Mathews Ave. two blocks north of Springfield Ave.
  • Midterm Exam 1: Thursday, 2/19, 7 pm - 8 pm, Room 1404 Siebel.
  • Midterm Exam 2: Thursday, 3/19, 7 pm - 8 pm, Room 1404 Siebel
  • Midterm Exam 3: Thursday, 4/30, 7 pm - 8 pm, Room 1404 Siebel
  These dates/times were chosen after polling the class; they minimize or eliminate conflicts with classes, other exams, and other academic functions. (In fact, at the time the poll was taken, only one student reported a legitimate conflict with any of the above dates.) Please mark your calendar and keep those dates free of other commitments. If you have a legitimate conflict at one of these dates, let me know immediately; in cases of a legitimate academic conflict (e.g., another class or lab or exam), I may provide a conflict slot, but you should first exhaust other options (e.g., in case of an evening exam for another class, take that class's conflict exam). All requests for conflict exams must be made in writing (ajh@uiuc.edu), with full details on the conflicting class/event, including exact start and end times, no later than one week before the date of the exam.
• [1/20/09] HW schedule: The first assignment will be given out Friday this week and will be due Friday next week (Jan. 30).

Handouts

• Central Limit Theorem
• Variance, covariance, correlation, and moment-generating functions
• Joint distributions
• Double integrasl: Tips and practice problems
• Solutions to practice problems on double integrals
• Important continuous distributions
• Continuous random variables.
• Important discrete distibutions
• Discrete random variables.
• Independence.
• Average Rule and Bayes' Rule.
• Conditional probabilities.
• Set-theoretic terminology and notations
• Binomial and multinomial coefficients

Computer simulations

• Central limit theorem demonstration. This is a java applet to demonstrate, in a quite dramatic fashion, the emergence of the normal distribution in sums of independent random variables. The applet allows you to specify the initial distribution by assigning probabilities for (up to) 10 values P(0), P(1), P(2), ..., P(9). The program then computes and plots, for given n, the distribution of a sum of n independent r.v.'s having this distribution. It is remarkable how quickly the distribution approaches the normal shape as one increases the value of n from n=1 (the original distribution) to 2, 3, etc. This occurs, sooner or later, no matter what the shape of the original distribution. For nice symmetric distributions (e.g., uniform) the convergence to the normal distribution is very fast; for other distributions it may take a bit longer to see the normal distribution emerge.
• Coin tossing simulation. This applet simulates a sequence of coin tosses and plots the accumulated "scores" (1 for a head, -1 for a tail).
• Buffon Needle simulation.
• Normal approximation to binomial distribution. A java applet that, for given values of n and p and a given range for k, plots the binomial distribution along with the approximating normal curve. The match is remarkably good, even if n is relatively small.
• Plot of the Binomial Distribution This java applet allows you to plot the binomial distribution for a given value of p and three values of n.
• Random Walk Simulation. Click on the "play" button (single arrow) to run the simulation. To make it more interesting, increase the value of n (number of trials/moves) by moving the slide ruler. Note that several values are marked by a red square: (1) the final value, representing your end location in the random walk interpretation, or total net winnings in the gambling interpretation; the places where the graph crosses the x-axis, representing times where you return to the initial location, or times where you are down to your initial capital.
• Simulation of the birthday problem:
  • Simulation I. A relatively simple, but nice java applet. From the Probability by surprise website at Stanford.
  • Simulation II. Much more sophisticated, with lots of bells and whistles. From the "Virtual Laboratories in Probability and Statistics" an the University of Alabama at Huntsville.
• Simulations for de Mere's problem:
  • Game I: roll a single die 4 times; you win if a six shows up
  • Game II: roll two dice 24 times; you win if a double six shows up.

Links

• Simeon Poisson (1781 - 1840), after whom the Poisson distribution and the Poisson approximation were named
• Reverend Thomas Bayes (1702 - 1761), of Bayes' rule fame
• Chevalier de Mere (1607 - 1684) (aka Antoine Gombaud), an amateur mathematician and gambler, whose questions about probabilities involving repeated rolls of dice (see below) led the birth of probability as a systematic mathematical theory.
• Andrey Kolmogorov (1903 - 1987), considered the father of modern probability, who put probability theory on a rigorous mathematical footing based on the "Kolmogorov axioms"

Class Diary

• Friday, 5/1/09: Class canceled because of yesterday's exam.
• Wednesday, 4/29/09: The Weak Law of Large Numbers. Markov and Chebychev inequalities.
  Read: Section 8.3.
• Monday, 4/27/09: Central Limit Theorem, continued.
  Handout: Central Limit Theorem
  Homework 11 (not graded): Chapter 8 (pp. 457--459), Problems 1, 4, 5, 6, 14, 15. This assignment will not be collected; I will pass out solutions next week.
• Friday, 4/24/09: The Central Limit Theorem.
  Read: Section 8.3.
• Wednesday, 4/22/09: Computing expectations via the indicator method.
  Read: Section 7.2 through Example 2j
• Monday, 4/20/09: Additional properties of independent r.v.'s. Moment-generating functions.
  
  Homework: HW 10, due Friday, 4/24/09.
• Friday, 4/17/09: Covariance and correlation. Definitions, properties, and intuitive interpretation.
  Read: The definitions and properties can be found in Section 7.4, though most the examples there are rather involved and well above exam level, and the text lacks examples and problems of more routine type. I plan to distribute a set of additional problems next week.
  Handout: Variance, covariance, correlation and moment-generating functions.
• Wednesday, 4/15/09: Question and Review session.
• Monday, 4/13/09: Conditional probabilities, continuous case.
  Handout: Joint distributions. Summary of definitions and formulas.
  Homework: HW 9, due Friday, 4/17/09.
• Friday, 3/10/09: More examples involving sums (and other combinations) of independent normal r.v.'s,
  Return to the discrete case. Matrix representation of joint p.m.f., with marginal p.m.f., appearing in the "margins". Conditional p.m.f.'s.
  Read: Section 6.4.
• Wednesday, 4/8/09: Distribution of quotients, products, and sums. Density of sums of independent r.v.'s. Sums of independent normal, binomial, Poisson r.v.'s.
  Read: Section 6.3
• Monday, 4/6/09: More examples and problems on joint distributions. Uniform distribution on a 2-dimensional region. Buffon's needle problem.
  Handout: Double integrals: Tips and practice problems
  Solutions to practice problems.
  Homework: HW 8, due Friday, 4/10/09.
• Friday, 4/3/09: Started Chapter 6. Joint Distributions, discrete and continuous case. Joint p.m.f.'s and joint p.d.f.'s. Computing probabilities via joint densities. Marginal densities. Independence of random variables.
  Read: Section 6.1 (all) and Section 6.2 (through Example 2c, inclusive), and do the Self-Test Problems 3 - 6 on p. 323. The latter problems, for which solutions are in the back of the book, are good additional examples for working with joint density functions.
  Review: Double integrals from Calculus III. Working with joint densities leads to double integrals, and you need to do know how to compute such integrals.
• Wednesday, 4/1/09: Wrapped up Chapter 5. Change of variables. Records, maxima/minima, of random variables.
• Monday, 3/30/09: Normal approximation to the binomial distribution. The continuity correction.
• Friday, 3/20/09: Class canceled because of yesterday's exam.
• Wednesday, 3/18/09: The normal distribution. Working with the Phi function and the normal table. Normal approximation to binomial distribution.
  Read: Section 5.4.
  Homework: HW 7, due 4/3/09. .
• Monday, 3/16/09: The uniform and exponential distributions. Memoryless property of the exponential distribution. Explanation of the waiting time paradox.
  Read: Sections 5.3, 5.5. Skip 5.5.1 (Hazard rates.)
  Handout: Important Continuous Distributions.
• Wednesday, 3/18/09: The normal distribution. Working with the Phi function and the normal table. Binomial approximation to normal distribution.
  Read: Section 5.4.
  Homework: HW 7, due 4/3/09. .
• Friday, 3/13/09: Started Chapter 5, Continuous Random Variables. Definition of a continuous random variable and differences to the discrete case. Probability density function (p.d.f.) and cumulative distribution function (c.d.f.) of a continuous r.v. Properties of p.d.f.'s and c.d.f.'s. Probabilities via p.d.f.'s. Expectation and variance.
  Read: Sections 5.1 - 5.2.
  Handout: Continuous Random Variables.
• Wednesday, 3/11/09: Chapter 4 recap/review.
• Monday, 3/9/09: The Poisson distribution. Poisson approximation to the binomial distribution.
  Read: Section 4.7 through p. 163. (You can skip the remainder of this section.)
  Homework: HW 6, due Friday, 3/13/09.
• Friday, 3/6/09: Computing expectations via the indicator method. The expected number of distinct birthdays in the birthday problem.
  Fun stuff:
  One-dimensional Random Walk Simulation. Click on the "play" button to run the simulation.
• Wednesday, 3/4/09: S/F trial examples, continued. Negative binomial and geometric distributions. The baseball world series problem, and problems of similar type (Banach match box problem, Laplace's "Problem of points"). Birthday problems.
  Read: Sections 4.8.1, 4.8.2, 4.8.3. Example 4j in 3.4.
  
  Handout: Discrete probability distributions.
• Monday, 3/2/09: Special probability distributions, overview. Independent success/failure trials (Bernoulli trials). Probability calculations within this model. Binomial distribution. Examples.
  Read: Section 4.6 through p. 157.
  Homework: HW 5, due Friday, 3/6/09.
• Friday, 2/27/09: Properties of expectation, expectation of a function of a random variable. Variance and standard deviation. The St. Petersburg paradox, and the "double stake if you lose" game.
  HW 5 Sneak Preview: Chapter 4, Problems 1, 4, 8(a)(c), 13, 15 and 16 (only compute the p.m.f.'s for the values 1 and 2), 20(a)(c), 21(b), 38. Most of these problems reduce to ordinary probability computations of the type that came up in earlier chapters and require techniques from those chapters.
  (The homework will be due Friday next week. The official HW 5 handout will be distributed on Monday; I may add a problem or two (or three) to the above list.)
• Wednesday, 2/25/09: Expectation. Definition and interpretations. Examples of random variables, constant r.v.'s, indicator r.v.'s, and computations of their p.m.f.'s and expectations.
  Read: Section 4.3, 4.4, 4.5. In 4.4 you can skip Examples 4b and 4c.
• Monday, 2/23/09: Started Chapter 4. Random variables, formal and informal definitions. Probability mass functions (p.m.f.'s) and cumulative distribution functions (c.d.f.'s).
  Read: Section 4.1, 4.2, 4.9. In 4.1 you can skip Example 1e.
  Handout: Discrete Random Variables.
• Friday, 2/20/09: Class canceled because of yesterday's exam.
• Wednesday, 2/18/09: Wrapped up independence and conditional probabilities, with a discussion of some pitfalls and some examples. De Mere's problem revisited.
  Read: Preread Section 4.1 through Example 1d. (Skip the final example, 1e.) In particular, familiarize yourself with the terminology and notation of a random variable. All of the examples in this section reduce to probability calculations of the type that came up in earlier chapters (and which could have been just as well given earlier), but recast using random variable notation and terminology.
• Monday, 2/16/09: Independence. Definition and motivation. Independence of more than two events. Independence of complements. Examples and applications.
  Read: Section 3.4. Among the examples in 3.4, focus on 4a, 4b, 4c, 4e, 4g. Example 4k (Gambler's Ruin problem) was mentioned last time as an example of the conditioning technique.
  Handout: Independence.
  Homework: HW 4, due Monday, 2/23/09.
• Friday, 2/13/09: More illustrations of conditional probabilities. The "conditioning" technique. The three biased coins problem. The Ballot Problem.
  Read: Example 4k (the gambler's ruin problem), p. 95. This is similar to the ballot problem, in that it illustrates "conditioning" to derive a recurrence relation for the probabilities sought. (The important part here is the derivation of the recurrence, given on top of p. 96. The solution of the recurrence given here is rather tedious, and not very illuminating.)
• Wednesday, 2/11/09: The Average Rule and Bayes' Rule. Applications: False positives. Polling data. Three biased coins.
  Handout 1: Bayes' Rule.
  Handout 2: Examples and Case Studies on Conditional Probabilities.
  Read: Section 3.3. The key definitions and formulas are collected in the handout below. Among the examples in 3.3, focus on the following: 3a, 3c, 3d, 3j, 3k, 3m. You can skip over the discussion of "odds" ratios (formula (3.3), p. 80).
• Monday, 2/9/09: Started Chapter 3. Conditional probabilities. Definition, properties. Interpretations: relative area in Venn diagrams; probability under additional information; probability relative to subpopulations. Examples: Tests with false positives/negatives; the three card trick. Translating word problems into mathematical language.
  Read: Section 3.2 and beginning of 3.5 (Prop. 8.1). Chapter 3 of the text is quite heavy on examples, of varying usefulness. Among the examples in 3.2, focus on 2b and 2f; the other examples in this section are less instructive, and many are easier done using combinatorial arguments as in Chapter 2. The key definitions and theorems are collected in the handout below.
  Handout: Conditional probabilities.
  Homework: HW 3, due Friday, 2/13/09.
• Friday, 2/6/09: Wrapped up Chapter 2 with applications of the inclusion/exclusion principle: permutations without fixed points and equivalent problems (hat check problem, matching problem, etc.), and counting words with required letters (e.g., 6 letter words formed with letters A,B,C containing all 3 of these letters), and equivalent problems (coupon collecting, birthdays, etc.).
  Read: Reread Section 2.4, especially the latter part on the inclusion/exclusion principle. (Section 2.5 was covered earlier. I will not cover the asterisk sections 2.6*, etc.)
  HW 3 Preview: The following problems, all from the Problem Section of Chapter 2 on p. 55 - 60, will be on HW 3: 1, 3, 5, 6, 10, 12, 13, 14. Also, Problem 6(a)-(d) on p. 61. These problems are on Sections 2.2 - 2.4. I will distribute an "official" HW assignment sheet, with more instructions/hints (and possibly an additional problem or two), on Monday.
• Wednesday, 2/4/09: Probability rules derived from Kolmogorov axioms. Working with axioms and rules. Translating verbal descriptions into set-theoretic expressions.
  Read: Section 2.4.
• Monday, 2/2/09: Tips on defining appropriate sample spaces. Examples where the equally likely outcome model breaks down. The Kolmogorov probability model. Kolmogorov axioms. Probability as an abstract function ("P-function") defined on subsets of S. Examples of P-functions satisfying the Kolmogorov axioms.
  Read: Sections 2.2/2.3.
  Handout: Set-theoretic terminology and notation.
  Homework: HW 2, due Friday, 2/6/09.
• Friday, 1/30/09: More birthday problems. Lottery and poker probabilities. The inclusion/exclusion principle (case of 2 and 3 sets).
  Read: For the inclusion/exclusion principle see p. 32-34 in 2.4 and illustrated by Examples 5l (for the case of three sets) and 5m (for the general case). You can focus on the special cases of two and three sets that were covered in class. The general case (as given in Prop. 4.4 on p.34) is more difficult, and will not be covered in exams. However, the most famous application of the inclusion/exclusion principle, the matching problem of Example 5m, requires the general case.
  HW 2 Preview: The following problems, all from the Problem Section of Chapter 2 on p. 55 - 60, will be on HW 2: 15, 16, 18, 23, 28, 35, 36, 37, 43, 44, 52. All involve probability computations of the type covered in Section 2.5 and in class on Wednesday and Friday this week. I will distribute an "official" HW assignment sheet, with more instructions/hints (and possibly an additional problem or two), on Monday.
• Wednesday, 1/28/09: The simplest probability model: the case of equally likely outcomes. More counting problems, this time disguised as probability questions. De Mere's problem. The birthday problem.
  Read: Section 2.5, especially Examples 5a, 5b, 5c, 5d, 5f, 5g, 5h. These are very instructive examples that illustrate counting arguments of the type that were developed in Chapter 1. (I intend to spend part of Friday doing more problems of this type, and, in particular, discuss poker probabilities.)
• Monday, 1/26/09: Sections 1.4/1.5. Combinations. Examples of problems involving combinations: Counting subsets, committee problems, lottery problems, box/ball type problems, HT sequences.
  Read: Sections 1.4/1.5.
  Handout: Binomial and multinomial coefficients.
• Friday, 1/23/09: More word counting problems and problems that can be modelled as word counting problems.
  Read: Sections 1.4 - 1.5.
  Homework: HW 1, due Friday, 1/30/09.
• Wednesday, 1/21/09: Started Chapter 1. Combinatorial analysis. Word counting problems of various types. Multiplication principle. Permutations.
  Read: Sections 1.1 - 1.3.