Introduction

Primary aim

The primary objective of this and the next several chapters is to introduce probability theory not as part of pure mathematics but as a mathematical framework for modeling certain observable phenomena which we call stochastic: phenomena that exhibit chance regularity (see chapter 1). Center stage in this modeling framework is given to the notion of a statistical model. This concept is particularly crucial in modeling observational (non-experimental) data. The approach adopted in this book is that the mathematical concepts underlying the notion of a statistical model are motivated by formalizing a generic simple stochastic phenomenon we call a random experiment. An example of such a phenomenon is that of “counting the number of calls arriving in a telephone exchange, over a certain period of time.” The formalization (mathematization) of this generic stochastic phenomenon will motivate the basic constituent elements that underlie the notion of a statistical model and provide the foundation for a broader framework in the context of which empirical modeling takes place.

Why do we care?

The first question we need to consider before we set out on the long journey to explore the theory of probability as a modeling framework is:

The answer in a nutshell is that it provides both the foundation and the frame of reference for data modeling and statistical inference.