Artificial intelligence

Abstract: In the paper, application of the principle of least action of mechanical systems to stochastic systems with continuous random variables is considered. Lagrange function is determined as a quadratic form according to distribution function F(x) and distribution density f (x)=F(x). As a result, differential equations, which lead to four well-known distribution laws of the theory of probability, i.e. random, linear, harmonic and exponential distribution, are obtained. The results were obtained in case when Lagrange function does not directly depend on the random variable, that’s why these results form the basis for further study of various forms of distributions.

Keywords: theory of probability, distribution function, distribution density, quadratic form, functional, action, parameter, stochastic systems, variational principle.

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