# Probability and Statistics

*Rob Carroll*

*2017-08-28*

# Introduction

These are the notes for POS 5737, taught in the Department of Political Science at Florida State University. They freely borrow from several well-known textbooks, including those by Wackerly, Mendenhall, and Scheaffer (2008), DeGroot and Schervish (2012), and Casella and Berger (2002). They also borrow from my own notes as a graduate student when I was taught by Kevin Clarke. Kevin was kind enough to provide his own old notes, which often lean on Spanos (1986). For the introduction of basic concepts—say, functions and their domains—I sometimes borrow from notes by Duggan (2013) and from the well-known introductory texts by Simon and Blume (1994) and Carter (2001). And, when I am feeling especially masochistic, I invoke the colossal Aliprantis and Border (2006).

These notes are meant to be self-contained and so do not begin with any prerequisite amount of knowledge in mind. There is no such thing as a typical political science graduate student, nor a typical background for entering a program like ours, and so the only requirement is dedication. The goal is to build up the basics of probability theory directly from set theory in the first two thirds of the course, and then to turn attention to statistics in the final third. The primary aim is to give students a coherent way to think about data—and models of data—that builds from axiomatic beginnings but that is not fully wedded to the analytic approach.

Students take this class concurrently with separate courses on research design and on mathematics. There are, of course, numerous overlaps, which is sometimes by design. But more subtly, there are times where the material presented here slows down a bit to make sure students have had the opportunity to learn relevant concepts from their other courses.

These notes have been brought together using the `gitbook`

option within the `bookdown`

package in `R`

, and so they seem to run together from chapter to chapter. But, this runs the risk of the reader inferring these are to be used like a textbook. They are not. These are my lecture notes, such as they are, and they should not be used for anything other than to reinforce what was said in class and to ensure that no notational mistakes persist long-term. The class’s assigned textbooks are to serve as the textbooks, which is the task they were designed to perform (to varying degrees of success, of course). The notes, on the other hand, serve the task of structuring lectures, as well as providing a unified notation.

Because the course notes have been compiled in `R`

, one may draw the inference that our approach will be computational. One would be wrong. Our approach will be *very* analytic by the discipline’s standards, with lectures involving old-fashioned pedagogy, a generally high level of rigor, and a fully axiomatic approach. Why, then, put the notes into `R`

? For starters, `RStudio`

provides a nice way to output notes into HTML. But, I will also use `R`

to provide visualizations that (hopefully) drive some of the more abstract ideas home. All code will be provided, but the student need not worry that exams or homeworks will involve anything other than a pencil and paper (and the occasional cold sweat).

The class is a work in progress, and so these notes should not be taken as a final product. In the current context, readers should be on the lookout for mistakes, which I have intentionally—and rather cleverly, might I add—sprinkled throughout to keep students on their toes. They should also point out things that are not as clear as possible. These mistakes refect the instructor’s own pedagogical and substantive limitations, but they also reflect ever-changing mathematics curricula in our own school system, along with different perspectives from students with varying degrees of mathematical sophistication.

### Probability Spaces

Wackerly, Dennis D., William Mendenhall III, and Richard L. Scheaffer. 2008. *Mathematical Statistics with Applications*. Seventh. Belmont, CA: Thomson.

DeGroot, Morris H., and Mark J. Schervish. 2012. *Probability and Statistics*. Fourth. Boston, MA: Addison-Wesley.

Casella, George, and Roger L. Berger. 2002. *Statistical Inference*. Second. Pacific Grove, CA: Duxbury.

Spanos, Aris. 1986. *Statistical Foundations of Econometric Modelling*. Cambridge, UK: Cambridge University Press.

Duggan, John. 2013. “Basic Concepts in Mathematical Analysis: A Tourist Brochure.” http://www.johnduggan.net.

Simon, Carl P., and Lawrence Blume. 1994. *Mathematics for Economists*. New York: W.W. Norton & Company.

Carter, Michael. 2001. *Foundations of Mathematical Economics*. Cambridge, MA: The MIT Press.

Aliprantis, Charalambos D., and Kim C. Border. 2006. *Infinite Dimensional Analysis: A Hitchhiker’s Guide*. Third. Berlin: Springer.