An Introduction to Probability and Statistics, 3rd Edition

6SAMPLE STATISTICS AND THEIR DISTRIBUTIONS

6.1 INTRODUCTION

In the preceding chapters we discussed fundamental ideas and techniques of probability theory. In this development we created a mathematical model of a random experiment by associating with it a sample space in which random events correspond to sets of a certain σ-field. The notion of probability defined on this σ-field corresponds to the notion of uncertainty in the outcome on any performance of the random experiment.

In this chapter we begin the study of some problems of mathematical statistics. The methods of probability theory learned in preceding chapters will be used extensively in this study.

Suppose that we seek information about some numerical characteristics of a collection of elements called a population. For reasons of time or cost we may not wish or be able to study each individual element of the population. Our object is to draw conclusions about the unknown population characteristics on the basis of information on some characteristics of a suitably selected sample. Formally, let X be a random variable which describes the population under investigation, and let F be the DF of X. There are two possibilities. Either X has a DF F_θ with a known functional form (except perhaps for the parameter θ, which may be a vector) or X has a DF F about which we know nothing (except perhaps that F is, say, absolutely continuous). In the former case let Θ be the set of possible values of the unknown parameter θ. Then ...