A numerical modeling and its computational implementing simulation for generating distributions of the complicated random variable transformations with applications

This paper presents a mathematical modeling of new numerical techniques which are used to approximate the probability distribution of the univariate random variable transformations, to overcome the restrictions that prevent applying common analytical methods in the most cases. The proposed study is implemented by deducing mathematical formulas that make it easy to determine the cumulative distribution function numerically for any transformation of a random variable at a point from its range. The effectiveness of employing this procedure depends on finding numerically the values of inverse transformation images of that point and since it is difficult to deal analytically with the transformation function, this research presents an alternative method that depends on using Taylor's expansion to find a best polynomial function that converges strongly to the transformation function on its domain which facilitates generating the inverse images. Mathematical algorithms are designed to illustrate the computational implementation process of all former procedures. A stochastic model related to the heat equation is considered to obtain the distribution of its probabilistic solution which has a complicated nature impedes applying the usual analytical methods, therefore applying this current study has become effective in such cases. The results have been validated using Mathematica software.