MATH 180A (Winter quarter 2008). Introduction to Probability

Instructor: David A. Meyer
Office hours: AP&M 7256, M 1:00pm-2:00pm, W 12:00nn-1:00pm, or by appointment
Lecture: University Center 413, room 2, MWF 9:00am-9:50am
Email: dmeyer "at" math "dot" ucsd "dot" edu

Teaching assistant: John Foley
Office hours: TBA
Section A01: Center Hall, room 205, Tu 9:00am-9:50am
Section A02: Center Hall, room 205, Tu 9:00am-9:50am
Email: jfoley "at" math "dot" ucsd "dot" edu

Course description

This course is an introduction to probability. The mathematical analysis of probabilities originated with attempts to optimize play in various gambling games, and probability continues to be a useful tool for describing many situations in the real world. In this course we will learn the basic ideas of both discrete and continuous probability.

Especially for the latter, the prerequisite for this course is a thorough knowledge of calculus. The textbook is Jim Pitman, Probability (New York: Springer-Verlag 1993). We will cover most of chapters 1-5, and possibly part of chapter 6.

There will be weekly homework assignments, due in section on Tuesdays, or before then in the drop box on the sixth floor of AP&M. Students are allowed to discuss the homework assignments among themselves, but are expected to turn in their own work — copying someone else's is not acceptable. Homework scores will contribute 15% to the final grade.

There will be two midterms, approximately in the fourth and eighth weeks of the quarter. The final is scheduled for 8:00am Wednesday 19 March. Scores on the two midterms and final will contribute 25%, 25% and 35% to the final grade, respectively. There will be no makeup tests.

Related events

25 Feb 08 Uri Treisman will speak "On innovation in urban mathematics education"
Price Center, Davis Room, 2:00pm-3:00pm
22 Feb 08 Carl Weiman will speak on "Science education in the 21st century: using the tools of science to teach science"
Natural Sciences Building (NSB) 1205, 12:00pm-1:00pm
15 Feb 08 Avi Wigderson will speak on "The power and weakness of computational randomness"
Computer Science and Engineering (CSE) building 1202, 11:00am-12:00pm
13-14 Feb 08 Informational meetings for CalIT2 Summer Undergraduate Research Scholarship Program
5th floor Atkinson Hall
8 Feb 08 Gerd Gigerenzer (see 18 Jan lecture) will speak on "Gut feelings: the intelligence of the unconscious"
Computer Science and Engineering (CSE) building 1202, 2:00pm-3:30pm

Syllabus (subject to modification)

7 Jan 08 §1.1. Equally likely outcomes
         flipping coins
         rolling dice
         drawing cards
         Las Vegas, odds
§1.2. Interpretations
         de Finetti's definition of probability [1]
         physics and symmetry
lecture notes
HWK (for W 9 Jan 08).
         Read §1.1-1.3.
9 Jan 08 §1.1. Equally likely outcomes
         outcome space, events
§1.3. Distributions
         Kolmogorov's axioms for probability functions [2]
         Venn diagrams [3,4]
§1.2. Interpretations [5]
         subjective, objective, frequency
         Extra Credit: Show that the frequency interpretation follows from the subjective interpretation.
         Hint: See [6].
         Iowa 2008 U.S. Presidential nomination markets
HWK (due Tu 15 Jan 08).
         §1.1: 3, 6, 7
         §1.2: 2, 3
         §1.3: 4, 6, 11
11 Jan 08 §1.3. Distributions
         calculating event probabilities
         histograms as probability functions
         Extra Credit: (Sam Buss) Which distribution functions are possible for a weighted die?
         special probability distributions
                 Bernouli(p)
                 discrete uniform
                 continuous uniform in 1 and 2 dimensions
         empirical distributions
                 air pollution data for Beijing
lecture notes
HWK (for M 14 Jan 08).
         Read §1.4-1.5.
14 Jan 08 §1.4. Conditional probability and independence
         de Finetti's definition of conditional probability [1]
         de Finetti's assumption of coherence of probability assignments [1]
         derivation of P(Ω) = 1 and P(AB) = P(A) + P(B) if AB = ∅
         Extra Credit: If the betting line is for the Patriots to beat the Chargers by 14 in the AFC Championship,
         what is the probability that the Chargers will win? More generally, if the betting line is $k$, what is the
         probability for the less favored team to win?
         Hint: Use an empirical probability distribution.
lecture notes
16 Jan 08          derivation of P(A|B) = P(AB)/P(B)
         idea of independence
lecture notes
HWK (due Tu 22 Jan 08).
         §1.4: 2, 4, 6, 11, 12
         §1.5: 2, 4, 5, 6
18 Jan 08          definition of independence of events
§1.5. Bayes' Rule [7]
         breast cancer example [8]
HWK (for W 23 Jan 08).
         Read §1.6, §2.1.
21 Jan 08 No lecture: MLK holiday
23 Jan 08 §1.6. Sequences of events
         infinite outcome spaces
         independence of preceding events
         dependence on preceding events
         birthdays problem
HWK (due Tu 29 Jan 08).
         §1.6: 2, 4, 5, 8
         §2.1: 2, 3, 6, 7, 8, 13, 14

25 Jan 08          mutual independence of multiple events
§2.1. The binomial distribution
         Theodore Postol's Congressional testimony on the effectiveness of the Patriot missile
28 Jan 08 §2.3. Normal approximation: derivation
         P(k) ∝ exp[-(k-μ)2/2σ2], where μ = pn and σ2 = npq
         normalize by setting integral to 1, which gives 1/σ√2π as the proportionality constant
§2.2. Normal approximation: method
         probability of being within 1, 2, or 3 standard deviations of the mean
29 Jan 08 Review session: University Center 413, room 1, 6:00pm-7:00pm
practice midterm; solutions
30 Jan 08 Midterm 1: §1.1-1.6, §2.1, lectures through 25 Jan
1 Feb 08          the standard normal distribution
         the cumulative distribution function
         Square Root Law
         Law of Large Numbers
§2.4. Poisson approximation
         derivation
HWK (due Tu 5 Feb 08).
         §2.2: 3, 9, 10, 13, 15
         §2.3: 2
         §2.4: 1, 5, 10
4 Feb 08 §2.5. Random sampling
6 Feb 08 No lecture
HWK (for F 8 Feb 08).
         Read §3.1-3.2.

HWK (due Tu 12 Feb 08).
         §2.5: 5, 6, 7, 10, 12
         §3.1: 4, 7, 11, 14, 16, 17, 22

8 Feb 08 §3.1. Introduction
         definition of random variable and its distribution
         Extra Credit: Prove that the distribution of a random variable is a probability distribution.
         joint and marginal distributions
         sum of random variables, convolution
11 Feb 08          letter frequencies, multinomial distribution
         Extra Credit: How could you use the difference between the multinomial distributions for
         draws of letters from English and Spanish text to automatically determine the language?
         Hint: Remember Bayes' rule.

§3.2. Expectation
         definition
         expectation value of function of random variable(s)
         E[X + Y] = E[X] + E[Y]
         X, Y independent implies E[XY] = E[X]E[Y]
13 Feb 08          expectation of binomial random variable, indicator random variable
         Markov's inequality
         prediction, loss function, risk
§3.3. Standard deviation and normal approximation
         definition of variance and standard deviation
         variance of binomial random variable
HWK (due Tu 19 Feb 08).
         §3.2: 3, 4, 8, 9, 14, 20
         §3.3: 2, 7, 9, 12, 13, 18

20 Feb 08 HWK (due Tu 26 Feb 08).
         §3.4: 4, 7, 10, 17, 20
         §3.5: 8, 9, 14, 18
         §6.4: 1, 3, 7, 10, 14, 18

26 Feb 08 Review session: University Center 413, room 1, 6:00pm-7:00pm
practice midterm; solutions
27 Feb 08 Midterm 2: §1.1-1.6, §2.1-2.5, §3.1-3.5, §6.4, lectures through 22 Jan
(emphasis on material covered since Midterm 1)

29 Feb 08 solutions to Midterm 2 problems
HWK (due Tu 4 Mar 08).
         §4.1: 2, 5, 6, 7, 12
         §4.2: 2, 4, 9, 12, 13

5 Mar 08 HWK (due Tu 11 Mar 08).
         §4.4: 2, 3, 6, 7, 9
         §4.5: 1, 3, 4, 6, 9

10 Mar 08 lecture notes
12 Mar 08 HWK (recommended).
         §5.1: 1, 3, 5, 9
         §5.2: 3, 5, 9, 13, 19
         §5.3: 3, 7, 11, 15

14 Mar 08 HWK (recommended).
         §6.5: 1, 2, 3

16 Mar 08 Review session: AP&M room B402A, 4:00pm
19 Mar 08 Final: §1.1-1.6; §2.1-2.2, 2.4-2.5; §3.1-3.5, §4.1-4.2,4.4-4.5; §5.1-5.3; §6.4-6.5
(emphasis on material covered since Midterm 2)

Suggested reading

[1] Bruno de Finetti, "La prévision: ses lois logiques, ses sources subjectives", Annales de l'Institute Henri Poincaré 7 no.1 (1937) 1-68;
translated from the French by Henry E. Kyberg, Jr., as "Foresight: its logical laws, its subjective sources", Samuel Kotz and Norman Lloyd Johnson, eds., Breakthroughs in Statistics. Vol.1: Foundations and Basic Theory (New York: Springer-Verlag 1992) 134-174.
[2] Andrei Nikolaevich Kolmogorov, Grundbegriffe der Wahrscheinlichkeitrechnung (1933);
translation edited by Nathan Morrison, Foundations of the Theory of Probability (New York: Chelsea 1956).
[3] John Venn, The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability (London: Macmillan 1876).
[4] Frank Ruskey and Mark Weston, "A survey of Venn diagrams", Electronic Journal of Combinatorics DS5 (2005).
[5] Alan Hájek, "Interpretations of probability", in Edward N. Zalta, ed., The Stanford Encyclopedia of Philosophy (Winter 2007 Edition).
[6] Edwin T. Jaynes, Probability Theory: The Logic of Science, edited by G. Larry Bretthorst (Cambridge: Cambridge University Press 2003).
[7] Thomas Bayes and Richard Price, "An essay towards solving a problem in the doctrine of chances", by the Late Rev. Mr. Bayes, F.R.S., communicated by Mr. Price, in a Letter to John Canton, A.M.F.R.S., Philosophical Transactions 53 (1763) 370-418.
[8] Gerd Gigerenzer, Calculated Risks: How to Know When Numbers Deceive You (New York: Simon & Schuster 2002);
Gerd Gigerenzer and Adrian Edwards, "Simple tools for understanding risks: from innumeracy to insight", British Medical Journal 327 (2003) 741-744.

Last modified: 16 Mar 08.