|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.
Class notes, quizzes, tests, homework assignments
- 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
common families of probability distributions, and the central limit theorem.
This course is cross-listed as Econ 360.
Clark’s Academic Catalog
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.
- 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
- 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
- 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
- Textbook. A First Course in Probability, 9th edition, by Sheldon Ross.
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
- 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
- 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
Assignment 1 due next Wednesday.
- Friday, Aug 29. Binomial coefficients.
Combinations, Pascal's triangle, multinomial coefficients, stars & bars, combinatorial proofs
The BinomialPlot and
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.
First assignment due.
- 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.
- Bayes' formula. Examples, tree diagrams
- Independent events. Definition, product spaces, independence
of more than two events, joint random variables, random samples, i.i.d. random variables
- The Bernoulli process. Sampling with replacement. Binomial
distribution, geometric distribution, negative binomial distribution, hypergeometric
distribution. Sampling without replacement
- Discrete random variables. Probability mass functions, cumulative
distribution functions. Various graphs and charts
- Expectation for discrete random variables. Definition,
expectation for the binomial and geometric distributions. St. Petersburg paradox.
Properties of expectation.
- Variance for discrete random variables. Definition and properties.
Variance of the binomial and geometric distributions.
- Continuous probability. Monte Carlo estimates. Introduction to
the Poisson process and the normal distribution. Statement of the central limit theorem.
- Density functions. Density as the derivative of the c.d.f., and
the c.d.f. asa the integral of density. Functions of random variables.
The Cauchy distribution.
- The Poisson process.
The Poisson, exponential, gamma, and beta distributions. Axioms for the Poisson process.
- Expectation and variance for continuous random variables.
Definitions and properties. Expectation and variance for uniform and continuous distributions.
Lack of expectation and variance for the Cauchy distribution.
- Moments and the moment generating function
- Joint distributions
- Sums and convolution
- Conditional distributions
- Covariance and correlation
- A proof of the central limit theorem
- A short introduction to Bayesian statistics
- Part I: an example, the basic principle, the Bernoulli process
- Part II: Bayes pool table, conjugate priors for the Bernoulli
process, point estimators, interval estimators
- Part III: the Poisson process & its conjugate priors
- Part IV: the normal distribution with known variance
- Part V: the normal distribution with unknown variance
This page is located on the web at
David E. Joyce