Interrelation of the concept of “uncertainty” in the information theory and theory of measurements

¹ “Sevastopol State University”, Sevastopol, Russian Federation
² “Tula State University”, Tula, Russian Federation

^* Corresponding author: AIBalakin@sevsu.ru

Abstract

The problem of determining the optimal number of multiple measurements based on the type of the error distribution density of the measuring means is considered. The laws of distribution of a random variable are obtained, they ensure the extreme values of dispersion at specified entropy. It is indicated that the given law corresponds to the known compositional one which ensures maximum entropy under the restrictions on variation limits of the random variable and at specified dispersion. In particular, under certain conditions, it corresponds to an abridged normal, uniform and bimodal laws. It is shown that two dispersion values correspond to a single entropy value. This effect is provided due to the fact that the random variable is concentrated on a finite interval. The theorem on the interrelation of entropy and dispersion of a random variable is proved which allows us to reconcile the concept of uncertainty used in the information theory with the same concept used in the up-to-date standards of technical measurements. It is shown that an abridged normal distribution law providing maximum of entropy at specified dispersion, at the same time provides the minimum of dispersion at specified entropy, and the bimodal law provides the maximum of entropy at the maximum of dispersion. The conclusions are based on the solution of two variational problems with isoperimetric constraints. The results of modeling allowing to evaluate the correctness of the conclusions are presented.