2025 , Volume 30, № 2, p.54-72

This paper examines both possibility and feasibility for replacing probability densities with their piecewise polynomial approximations as formulae, which are implemented for the inverse distribution method are expensive in computer realization.

It is shown that the use of high-degree polynomials is inappropriate from the point of view of constructing the corresponding algorithm of the discrete superposition method (due to the presence of negative coefficients in such expansions), and the best recommendation is the use of piecewise linear or piecewise constant density approximations. The results of numerical experiments (for a power-law distribution) are presented, which show that the cost of implementing the modified discrete superposition method when using a piecewise constant density approximation is slightly less than for the piecewise linear approximation.

At the sametime, the work predicts upper boundaries for the errors of the normalized approxima tions. In this case, the estimates for the piecewise linear approximation are less than for the case of piecewise constant approximation by an order of magnitude (in terms of the uniform grid step). This difference in the quality of approximation is clearly demonstrated by the example of the same power-law distribution. It justifies the recommendation formulated in the work on the advisability of preferential use of piecewise linear approximations of densities.

Finally, the work shows that for piecewise linear and piecewise constant approximations allow using a new technique — approximate probability equalization, which means constructing a partition of a finite interval of the distribution of a random variable for which the probabilities of falling into the half-intervals of the division are equal. This allows radical increasing the efficiency (cost effectiveness) of the computer modelling for a random variable with respect to approximation of its density (the work presents the results for experiments of corresponding indicative computer simulations).

Taking into account a certain distortion of the simulated distribution when using piecewise constant and piecewise-linear approximations and the considerations given in the work for the probability equalization, we can note the prospects of using the double-sided rejection method with piecewise-constant majorant and minorant for probability densities with labor-intensive modelling formulas of the inverse distribution function method. The last remark requires a separate detailed study.