Overview

We now begin three chapters which are primarily aimed at a discussion of the main concepts of frequentist statistical inference. This is currently the prevailing approach to much of scientific inference, so a student should understand the main ideas to appreciate current literature and understand the strengths and limitations of this approach.

In this chapter, we introduce the concept of a random variable and discuss some general properties of probability distributions before focusing on a selection of important sampling distributions and their relationships. We also introduce the very important Central Limit Theorem in Section 5.9 and examine this from a Bayesian viewpoint in Section 5.10. The chapter concludes with the topic of how to generate pseudo-random numbers of any desired distribution, which plays an important role in Monte Carlo simulations.

In Chapter 6, we address the question of what is a statistic and give some common important examples. We also consider the meaning of a frequentist confidence interval for expressing the uncertainty in parameter values. The reader should be aware that study of different statistics is a very big field which we only touch on in this book. Some other topics normally covered in a statistics course like the fitting of models to data are treated from a Bayesian viewpoint in later chapters.

Finally, Chapter 7 concludes our brief summary of frequentist statistical inference with the important topic of frequentist hypothesis testing and discusses an important limitation known as the optional stopping problem.