Infinitesimal Parametrical Families of Distributions and the Statistical Foundations of Quantum Mechanics

Parametrical families of distributions are considered, characterised by Borel's measure, infinitisimal in parameters space. The latter property defines the construction of the family and each its member. The properties of this construction are studied, as well as techniques of synthesis of probabilistic distributions, and application areas, first of all quantum systems. They are defined as infinitisimal dynamic systems, observation results of which are described by above mentioned families with the wave function as a parameter. We have demonstrated that axiomatic system of the quantum mechanics is overdefined ("redundant" axioms are restated as theorems), as well as showed new statistical properties of relativistic quantum mechanics. We also proposed non- contradictive model to explain some paradoxal properties of quantum field (paradox of the boson field energy).