Chapter 6

Statistical Inference

# 6.1 Introduction

In the previous chapters, we developed a theory of probability that allows us to model and analyze random phenomena in terms of random variables and their distributions. While developing this theory, we often referred to real-world observations and data sets, for example, in the assumption that the tropical cyclones in Example 2.36 follow a Poisson distribution with mean 15. Although we might be able to argue that the distribution should be Poisson from purely physical and meteorological principles point of view, where did the number 15 come from? It is simply the average number of cyclones per year that has been observed during the years 1988–2003, so we used this measured value as our parameter. This is a typical situation in any application of probability theory. We formulate a model by making assumptions about distributions and their parameters, but in order to be able to draw any useful conclusion, we need data. In this chapter, we outline the field of statistical inference, or statistics for short, which ties together probability models and data collection.

# 6.2 Point Estimators

Suppose that we manufacture lightbulbs and want to state the average lifetime on the box. Let us say that we have the following five observed lifetimes (in hours):

983, 1063, 1241, 1040, 1103

which give us the average 1086. If this is all the information we have, it seems reasonable to state 1086 as the average lifetime (although “at least 1000 h” might ...