Week starting
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Monday
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Wednesday
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Friday
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Homework
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topic planned
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notes
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4 Sept |
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Ex 1, Ex 2 |
Ex 3; Ex 4 up to
state diagram |
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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
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4 Dec |
Ex 59 |
Ex 60 |
Ex 61 |
#10 due 6 Dec |
Brownian motion |
Chap 13
Ex 59-61
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11 Dec |
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Friday 15 Dec
FINAL EXAM 2:00 to 5:00 ML 211 |
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Syllabus
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
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Probability rules (P1) through (P5). Examples: three coins; four of a kind; prisoner's dilemma.
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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.
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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.
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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.
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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.
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Continuous distributions, introduced via the Binomial/beta relationship. Much discussion of approximation.
Definition of probability densities. Expected value formulae, derived by conditioning formula.
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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.
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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.
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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.
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Bivariate densities defined. Jacobians by first principles. Examples: linear combinations of normals.
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Example: betas from independent gammas. Marginal densities. Mention of chi-square. Conditional densities, by first principles.
Fall recess
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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.
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Generating functions. Examples: Poisson; negative binomial; gamma mixtures of Poissons;
Poisson approximation to Binomial; branching processes (big polynomials from Mathematica). Wind down.