THE ISSUE OF FINDING THE MAIN STATISTICAL FUNCTIONS BY DIRECT CALCULATION IN THE TEACHING OF THE DISCIPLINES "PROBABILITY THEORY" AND "MATHEMATICAL STATISTICS"

Abstract

Summary. Introduction. For students’ qualitative mathematical training, it is crucial to carefully and consistently present theoretical material. This ensures that students develop logical and continuous understanding of theoretical problems within a discipline, as well as the principles of their solutions from the beginning to the end of the course.

In the teaching of «The Theory of Probability and Mathematical Statistics», we have identified a gap in this logical chain – specifically in the determination of the basic statistical functions of Pearson, Student, and Fisher through the Euler gamma function. Typically, neither the gamma function itself, nor its properties, nor its applications are included in students’ mathematical training. This omission creates difficulties in understanding statistical functions at a deeper level.

Purpose. The purpose of this study is to present an alternative approach to deriving statistical functions without relying on special functions such as the gamma function. By using fundamental differentiation and integration methods, we aim to provide a more accessible approach for students. This approach facilitates understanding statistical function derivations by constructing them sequentially, step by step.

Methods. To construct the formulas for the statistical functions, we employ a stepwise convolution method. For finding the formula of the Pearson distribution, we use the formula for the convolution of two densities c². First, the convolution was applied to the degrees of freedom k₁=1 and k₂=1, which allowed to be determined c² for k = 2. Then for the pair k₁=1 and k₂=2 we obtain c² with degrees k = 3 and k = 4, and then, consistently, we match c² with k, equal to 5, 6, 7, 8, and so on.

To find the density of the Student’s distribution, we use a somewhat more complicated computation procedure, since it is a function of the distribution of the particle density, and the density of the divisor must be calculated each time separately. Similarly, you can find the Fisher function.

To verify the accuracy of our results, we compare our derived formulas with those obtained using general expressions based on the gamma function.

Results. The comparison between our derived formulas and the standard gamma function-based formulas shows complete correspondence. This confirms the validity of our proposed approach. Our method provides an intuitive, step-by-step construction of statistical function formulas without requiring prior knowledge of the gamma function.

Originality. This study presents a new pedagogical approach to teaching statistical functions in probability theory and mathematical statistics. By avoiding reliance on special functions, we make the learning process more accessible to students who may lack prior exposure to the Euler gamma function. Our stepwise approach enhances comprehension and supports deeper engagement with the mathematical structures underlying statistical distributions.

Conclusion. The proposed approach does not seek to replace the existing optimal computation methods using the gamma function. Rather, it serves as a pedagogical tool to bridge the gap between fundamental definitions of statistical functions and their more complex formulations.

Furthermore, the study expands the range of exercises and problems available in the section «Functions of a Random Argument» in mathematical statistics. This contributes to better assimilation of theoretical concepts and reinforces students’ understanding of statistical function derivations.