Mathematical Statistics and Stochastic Processes

Chapter 1

Introduction to Mathematical Statistics

1.1. Generalities

We may define statistics as the set of methods that allow, from the observation of a random phenomenon, the obtainment of information about the probability associated with this phenomenon.

It is important to note that the random character that we attribute to the considered phenomenon is often merely a way to translate our ignorance of the laws that govern it. Also, a preliminary study, taking into account only the observations made, proves interesting; this is the aim of data analysis.

Data analysis is the set of techniques of statistical description whose purpose is the treatment of data without any probabilistic hypotheses. It aims, in particular, to reveal the dominant parameters among those upon which the observation depends.

Descriptive statistics also treats observations without formulating any prior hypotheses, but we may consider these hypotheses to be underlying, since it essentially consists of studying the empirical probabilistic characteristics of observations (histograms, empirical moments, etc.).

However, we may include data analysis in descriptive statistics by defining it as the set of methods that allow the ordering of data and their presentation in a usable form.

Furthermore, simulation is an additional technique often used in statistics: it consists of carrying out fictitious experiments in such a way as to make visible the expressions of chance in the evolution of a phenomenon. A simple, important ...