Statistics 241/541, Fall 2000

Week starting Monday Wednesday Friday Homework topic planned notes
4 Sept   Ex 1, Ex 2 Ex 3; Ex 4 up to state diagram   Probability rules. Chap 1,
Ex 1--6
11 Sept Ex 4 up to 5/12 soln; Ex 5 Ex 6 (briefly); Ex 9 A (E5) and Ex 9B; Ex 8; heuristics for Ex 10 #1 due 13 Sept Conditioning. Start expectations.
18 Sept Ex 10; Ex 11; final symmetry argument for Ex 11; Ex 12. Ex 13, with discussion of (E5) #2 due 20 Sept (conditional) expectations. Chap 2  Ex 7--14
25 Sept Ex 15, 16, 17. Ex 18; state Ex 20. Ex 20; discussion of ways to find symmetry #3 due 27 Sept Binomial distribution. Start symmetry. Chap 3  Ex 15--19
2 Oct Explanation for Ex 21; start variances Ex 23 (only first part); Ex 24; Ex 25. Ex 26; sketch of Ex 27 #4 due 4 Oct Symmetry, sampling, variances, covariances. Chap 4   Ex20--22
Chap 5   Ex23--27
9 Oct Ex 28 Ex 29 Ex 30, 31,32 #5 due 11 Oct Continuous distributions and densities. Chap 6  Ex 28--32
16 Oct Ex 33, 34 Ex 35, 36 MIDTERM test
(in class)
#6 due 18 Oct Normal distributions and CLT. Chap 7  Ex 33-38 Appendix
23 Oct Ex 37, 38 Ex 39, 40 Ex 41;
start Poisson proc
no homework due this week Poisson approximation and Poisson processes. Chap 8
Ex 39-41
30 Oct Ex 42; std exp;
indep times between points
Ex 43, 44 Ex 45, 46 #7 due 1 Nov Gamma and other distributions derived from Poisson processes. Chap 9
Ex 42-46
6 Nov Ex 47; start Ex 48 finish Ex 48; mention Ex 49; start Ex 50 finish Ex 50; 51 #8 due 8 Nov Bivariate densities. Jacobians by first principles. Chap 10
Ex 47-51
13 Nov DP Jury duty;
no lecture at Yale
start Ex 52 finish Ex 52; Ex 53 #9 due 15 Nov More bivariate densities: conditional densities. Chap 11
Ex 52-53
20 Nov Fall recess---no classes
27 Nov Ex 54 Ex 55, 56 Ex 57, 58 no homework due this week Multivariate normal. Chap 12
Ex 54-58
4 Dec Ex 59 Ex 60 Ex 61 #10 due 6 Dec Brownian motion Chap 13
Ex 59-61
11 Dec   Friday 15 Dec
2:00 to 5:00
ML 211


I expect to cover the material at about the same pace as in 1997 (see summary below). Of course, the addition of new material and the deletion of some old topics will change things slightly. I reserve the right to rearrange topics slightly to accommodate the needs and interests of the class.

Short description

Probability theory gives a systematic method for describing randomness and uncertainty. This course will explain the rules for manipulating random variables, probabilities, and expectations, with emphasis on the role of conditioning. The theory will be presented and motivated through a sequence of applications, ranging from the (traditional, boring) calculation of probabilities for card games to (more involved, more interesting) stochastic models.

The coin tossing model will generate the standard discrete distributions Binomial, Poisson, geometric, negative binomial. The Poisson process, the continuous time analog of coin tossing, will generate the standard continuous distributions exponential and gamma.

Normal approximations and calculations related to the multivariate normal distribution will exercise the multivariable calculus skills of the class (or provide a crash course in multiple integrals).

Applications to include: Markov chains; the probability theory of games, gambling, and insurance; coding theory; queueing theory; branching processes; geometric probability and stereology; (maybe) analysis of algorithms.

Outline of topics covered in 1997, numbered by week

  1. Probability rules (P1) through (P5). Examples: three coins; four of a kind; prisoner's dilemma.

  2. Conditioning. Conditional independence. Random variables and their distributions. geometric(p) distribution. Examples: HHH vs. TTHH; three way duel. Start on expectations and rules (E1) through (E4). Example: expected value for HHHH vs. TTHH.

  3. Practice with expectations. Examples: expected value of geometric; balls from urn (with extended discussion of strategy for solving such problems); brief mention of big pills, little pills. Binomial coefficients, reminder about Taylor expansion, binomial distribution. Example: expected value for binomial distribution by two methods.

  4. Examples: mugging and the binomial distribution; Bayes's formula (with small detour into Bayesian inference and confidence intervals). Variances and covariances defined. Uncorrelated versus independent. Tchebychev's inequality.

  5. Variance as a measure of concentration. Example: sampling (mention of Census Bureau's long form); variances by conditioning (not stressed enough). Unexpected symmetry. Examples: sampling without replacement from an urn (with much discussion of why there is symmetry); bet red, with impromptu detour into the ballot theorem.

  6. Continuous distributions, introduced via the Binomial/beta relationship. Much discussion of approximation. Definition of probability densities. Expected value formulae, derived by conditioning formula.

  7. Normal distributions. Density function. Graphical illustration of de Moivre's result. Example: normal approximation and summonses of Hispanic jurors. The central limit theorem, informally. Example: boxplots. Brief discussion of Stirling's formula, the \sqrt{2\pi} constant, and how one might derive de Moivre's result from the Binomial/beta correspondence.

  8. Poisson approximations. Mean and variance of Poisson. Example: letters in envelopes, hint of Poissonness; brief discussion of Stein's method (too hard). Poisson processes as very fast coin tossing. Example: gamma density from the Poisson process.

  9. Gamma functions and gamma densities. Means and variances. Exponential as special case. Example: square of N(0,1) as a gamma, introduction to method for calculating densities. Conditioning on events with zero probability. Example: process of arrivals, reason for appearance of Poisson processes. Analogies between distributions derived from coin tossing and distributions derived from Poisson process.

  10. Bivariate densities defined. Jacobians by first principles. Examples: linear combinations of normals.

  11. Example: betas from independent gammas. Marginal densities. Mention of chi-square. Conditional densities, by first principles.

    Fall recess

  12. Multivariate normal distribution, construction via correlation from independent standard normals. Regression to the mean. Brief mention of symmetry of multivariate normal density, and its consequences.

  13. Generating functions. Examples: Poisson; negative binomial; gamma mixtures of Poissons; Poisson approximation to Binomial; branching processes (big polynomials from Mathematica). Wind down.