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
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.
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 5^{th} 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 |
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(A∪B) = P(A) + P(B) if A∩B = ∅ 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(A∩B)/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}/2σ^{2}], where μ = pn and σ^{2} = 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) |
[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. |