Supplementary mathematics/Statistics and Probability

Statistics, in the popular sense of the term, deals with the group study of a population with the help of mathematics. In descriptive statistics, we simply describe a sample using quantities such as the mean, the median, the standard deviation, the proportion, the correlation, etc. This is often the technique that is used in censuses.

In a broader sense, statistical theory is used in research for inferential purposes. The goal of statistical inference is to draw the portrait of a given population, from the more or less blurred image formed using a sample from this population.

In another order of ideas, there is also “mathematical” statistics where the challenge is to find judicious (unbiased and efficient) estimators. The analysis of the mathematical properties of these estimators is at the heart of the work of the mathematician specializing in statistics.

The terms inductive statistics, appraisal statistics and inferential statistics ( inferential statistics ) are mostly used synonymously , which characterize the part of the statistics that is complementary to the descriptive statistics. Together with the theory of probability, mathematical statistics form the branch of mathematics known as stochastics .The mathematical basis of mathematical statistics is the theory of probability.

Probability theory in mathematics is the study of phenomena characterized by chance and uncertainty. Along with statistics, it forms the two sciences of chance which are an integral part of mathematics. The beginnings of the study of probabilities correspond to the first observations of chance in games or in climatic phenomena, for example.

Bell curve, histogram and dice.

Although the calculation of probabilities on questions related to chance has existed for a long time, the mathematical formalization is only recent. It dates from the beginning of the 20th century with Kolmogorov  's axiomatics. Objects such as events, probability measures , probability spaces or random variables are central in the theory. They make it possible to abstractly translate behaviors or measured quantities that can be assumed to be random. Depending on the number of possible values ​​for the random phenomenon studied, probability theory is said to be discrete or continuous.. In the discrete case, that is to say for at most a countable number of possible states, the theory of probabilities approaches the theory of enumeration  ; whereas in the continuous case, the theory of integration and the theory of measure provide the necessary tools.

Probabilistic objects and results are a necessary support for statistics, this is the case for example of Bayes' theorem , the evaluation of quantiles or the central limit theorem and the normal law. This modeling of chance also makes it possible to resolve several probabilistic paradoxes.

Whether discrete or continuous, stochastic calculus is the study of random phenomena that depend on time. The notion of stochastic integral and stochastic differential equation are part of this branch of probability theory. These random processes make it possible to make links with several more applied fields such as financial mathematics, statistical mechanics , image processing , etc.