Random processes

The article brings together a few elements of probability theory with a view to subsequent applications to physical data processing. In this context, the title "random processes" means that we are dealing with systems governed by a large number of parameters, providing one or more variables with unpredictable behavior.

On several occasions in the text, the terms "stochastic" and "statistical" will be used, sometimes in place of the adjective "random". Strictly speaking, these three terms do not necessarily have similar properties. While the term "random" is frequently used to designate variables that have no deterministic character, this is not the case for variables in the stochastic or statistical sense.

For example, today's weather forecasts provide daily proof that average temperatures in a given geographical location at any given time of year are perfectly predictable. On the other hand, the actual temperature estimated over the long term remains randomly situated around the average. For this reason, the temperature variable has the properties of a stochastic variable. As for the term "statistics", its use is generally confined to the construction of databases fed by random variables. It is also used to describe the criteria attached to these data. For example, the margin of uncertainty generated by calculating the mean value of a population of N random variables is based on statistical properties.

The article is divided into six sections, each of which is devoted in turn to the definition of events and random variables, the probabilities of discrete random variables, the probability densities of continuous random variables, the calculation of averages and moments, the usual laws of probability and, finally, the extension of probability theory to two or more random variables.

The various subdivisions of the six paragraphs are enhanced by examples taken mainly from the kinetic theory of gases. Indeed, the link between the microscopic mechanics of molecules and the thermodynamic concept provides an ideal case study for the field of application currently under consideration.

This text and its natural extensions, followed by the articles [R 220] and [R 221]...