The Set of Basis Functions Generated by Pearson Type IV Distributions and Its Application to Problems of Statistical Data Analysis and Quantum Mechanics

Using an example of Pearson type IV distributions, we propose a procedure of completing the classical probability distribution to a quantum state. We obtain a wave function corresponding to Pearson type IV distributions and construct the corresponding set of basis functions. Then we demonstrate how the developed method applies to problems of statistical data analysis and quantum mechanics, and show the efficiency of our approach for the problem of approximating statistical distributions with heavy tails.