Statistical testing of random variables

Statistical analysis is used today in scientific fields as varied as applied mathematics, physics, chemistry, economics and many others. This article introduces the reader to statistical tests adapted to random variables established according to the most familiar laws of probability. It goes without saying that the subject can be explored in greater depth by consulting the articles [AF 168] and [AF 170] in Techniques de l'Ingénieur or the specialized literature, some of which is listed at the end of the text [Doc. R 221].

The presentation is divided into three sections, dealing with three families of problems: estimating averages, statistical testing and simulating experiments involving artificial random variables.

The first paragraph highlights the impact of the uncertainty encountered when calculating the mean value or variance of N variables. We will demonstrate, with examples, that uncertainty can be estimated using the law of large numbers coordinated by the Bienaymé Chebetchev inequality. We will also observe the natural tendency of estimators to evolve towards the normal probability distribution.

The second paragraph focuses on the comparison of populations of random variables with known probability laws. The problem is to find criteria for measuring the deviation of a histogram of probability densities, or distributions, from a theoretical law taken from the catalog. The criteria in question will be based on the calculation of statistical gauges, the transformation of which will enable us to decide whether to reject or accept the candidate probability law. This important part will be illustrated by setting up the χ ² statistical test and the Kolmogorov Smirnov test. The second paragraph also deals with random variables expressed as complex numbers. Assuming that the real and imaginary components of these variables evolve according to the normal probability law, it will be shown that they can generate exponential, Rayleigh or Weibull probability laws.

The third paragraph deals with the simulation of physical systems through the generation of random variables produced by Monte Carlo draws. At this point, we briefly review the arithmetic congruence methods used in the architecture of algorithms for generating random numerical sequences. The analysis will continue with the development of examples...