MATH 461, Sections B/C, Spring 2009
Final Exam and Course Grades
Final Exam scores and course grades are now online and can be accessed as
usual via
this link.
The maximal score on the final was 290 + 10 EC points,
the highest score was 300, the average and median 194.
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:
Have a good break and enjoy your summer!
Course Policies, Exams, Grades
 Course Information Sheet.
The firstday handout. Everything you need to know about this class:
Syllabus, grading policies, office hours, etc.

Online Scores.
Log in with your NetID 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

Midterm Exam 2: Thursday, 3/19, 7 pm  8 pm, Room 1404 Siebel.

Midterm Exam 1: Thursday, 2/19, 7 pm  8 pm, Room 1404 Siebel.
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
Computer simulations
The links below are to java applets that simulate various probability
experiments. For the applets to run, you need to have
java installed. If you have problems running these applications, use
one of the computers in the Math library, or in one of the University
computer labs. (I tried it in the Math library, and
it worked fine there.)

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 xaxis, representing times where
you return to the initial location, or times where you are down to your
initial capital.

Simulation of the birthday problem:
 Simulations for de Mere's problem:
Links
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. 457459), 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.
Momentgenerating 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 momentgenerating 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 2dimensional 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 SelfTest 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:
Onedimensional 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 settheoretic 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 ("Pfunction") defined
on subsets of S. Examples of Pfunctions satisfying the Kolmogorov axioms.
Read: Sections 2.2/2.3.
Handout: Settheoretic 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. 3234 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.
Last modified: Sun 21 Feb 2010 12:56:13 PM CST
A.J. Hildebrand