# Chapter 8Statistical Inference

We conclude the probability part of this book with a chapter dedicated to statistical applications. This chapter is not meant to replace a detailed statistical book; instead, we present some common methods that are very useful for applications and for stochastic processes. This chapter is essentially dedicated to parameter estimation methods. For a study of the relationship between two or more variables, we direct the reader to a book such as Kutner, Nachtsheim, Neter, and Li (2004).

## 8.1 The Classical Problems in Statistics

In general, in statistics we work with a random sample. We recall here the definition 7.36 on page 233 of a simple random sample.

Suppose that we work on a probability space . A **simple random sample** of size *n* is a set of independent identically distributed (i.i.d.) random variables . The outcome of the sample, or simply a sample, is a set of realizations of these random variables. This random sample is called *simple* because the components are independent. Obviously, in this case the joint distribution is written as the product of marginal distributions. Typically, in statistics all the random variables in the sample have a distribution ...