Math 217, Probability and Statistics
Fall 2014 Prof. D. Joyce, BP 322, 793-7421 Department of Mathematics and Computer Science Clark University |

Pierre-Simon Laplace (1749–1827) | |||
---|---|---|---|

Portrait of Laplace at the Palace of Versailles
Date 1842, posthumous portrait by Sophie Feytaud |

Please bookmark this page, http://aleph0.clarku.edu/~djoyce/ma217/, so you can readily access it.

Course information

**General description.**An introduction to probability theory and mathematical statistics that emphasizes the probabilistic foundations required to understand probability models and statistical methods. Topics covered will include the probability axioms, basic combinatorics, discrete and continuous random variables, probability distributions, mathematical expectation, common families of probability distributions, and the central limit theorem.

This course is cross-listed as Econ 360.

See also Clark’s*Academic Catalog***Prerequisites.**Math 130 Linear Algebra, and Math 131 Multivariate Calculus**Course goals.**- To provide students with a good understanding of the theory of probability, both discrete and continuous, including some combinatorics, a variety of useful distributions, expectation and variance, analysis of sample statistics, and central limit theorems, as described in the syllabus.
- To help students develop the ability to solve problems using probability.
- To introduce students to some of the basic methods of statistics and prepare them for further study in statistics.
- To develop abstract and critical reasoning by studying logical proofs and the axiomatic method as applied to basic probability.
- To make connections between probability and other branches of mathematics, and to see some of the history of probability.

**Syllabus**- Basic combinatorics. Additive and multiplicative principles, permutations, combinations, binomial coefficients and Pascal’s triangle, multinomial coefficients
- Kolmogorov’s axioms of probability. Events, outcomes, sample spaces, basic properties of probability. Finite uniform probabilities. Philosophies of probability.
- Conditional probability. Bayes’ formula, independent events, Markov chains
- Random variables. Discrete random variables/distributions, expectation, variance, Bernoulli and binomial distributions, geometric distribution, negative binomial distribution, expectation of a sum, cumulative distribution functions
- Continuous random variables. Their expectation and variance. Uniform continuous distributions, normal distributions, Poisson processes, exponential distributions; gamma, Weibull, Cauchy, and beta distributions
- Joint random variables. Their distributions, independent random variables, and their sums. Conditional distributions both discrete and continuous, order statistics
- Expectation. Of sums, sample mean, of various distributions, moments, covariance and correlation, conditional expectation
- Limit theorems. Chebyshev’s inequality, law of large numbers, central limit theorem

**Textbook.***A First Course in Probability,*9th edition, by Sheldon Ross. Pearson, 2014.

We may occasionally also refer to an on-line textbook, Charles M. Grinstead and J. Laurie Snell’s*Introduction to Probability*, published by the American Mathematical Society, 1997, on-line at http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html ..**Course Hours.**MWF 9:00–9:50.**Assignments & tests.**There will be numerous short homework assignments, mostly from the text, occasional quizzes, two tests during the semester, and a two-hour final exam during finals week.**Time and study.**Besides the time for classes, you’ll spend time on reading the text, doing the assignments, and studying of for quizzes and tests. That comes to about five to nine hours outside of class on average per week, the actual amount varying from week to week.**Course grade.**[tentative] The course grade will be determined as follows: 2/9 assignments and quizzes, 2/9 each of the two midterms, and 1/3 for the final exam.

The dates for the discussion topics and the assignments are tentative. They will change as the course progresses.

- [to be filled in as the course progresses]
- Monday, Aug 25. Welcome and introduction to the class

Intro to probability via discrete uniform probabilities. Symmetry. Frequency.

Simulations and random walks - Wednesday, Aug 27.

Background on sets. Unions, intersections, complements, distributivity, DeMorgan's laws. Product, power sets. Countable and uncountable infinities.

Combinatorics. Principle of inclusion and exclusion, multiplicative principle, permutations, factorials and Sterling's approximation

Permutation generating applet - Friday, Aug 29. Binomial coefficients.

Combinations, Pascal's triangle, multinomial coefficients, stars & bars, combinatorial proofs

The BinomialPlot and GaltonBoard applets

Monday, Sep 1. Labor Day. No classes - Wednesday, Sep 3. Axioms for probability distributions.

Probability mass mass functions for discrete distributions, density functions for continuous distributions. Cumulative distribution functions. Sample spaces, axioms, and properties that follow from the axioms. discrete and continuous - Friday, Sep 5. Uniform finite probabilities

Odds. Repeated trials. Sampling with replacement. The birthday problem.

The Birthday applet - Monday, Sep 8.
Conditional probability. Xox, definition of conditional
probability, the multiplication rule
- Wednesday, Sep 10.
- Friday, Sep 12.

- Common probability distributions

Table of distributions - Java applets at http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/bookapplets/index.html

This page is located on the web at

http://aleph0.clarku.edu/~djoyce/ma217/