Duality of the Principle of the Entropy Maximum

The compositional distribution law of a random variable is obtained, which ensures a minimum of dispersion at a given entropy, as well as the condition that the distribution density at the boundaries of the interval of existence is zero. It is shown that this law complies with the known composition, which provides the maximum entropy with restrictions on the limits of variation of a random variable and for a given dispersion. It is shown that the obtained compositional distribution law, which ensures the minimum of dispersion at a given entropy, completely coincides with the composition law, which ensures the maximum entropy at a given dispersion. The conclusions are based on the solution of a variational problem with isoperimetric constraints. The simulation results are given, allowing to evaluate the correctness of the conclusions made.