ФОРМУЛА ОБРАЩЕНИЯ ДЛЯ РАЦИОНАЛЬНЫХ ХАРАКТЕРИСТИЧЕСКИХ ФУНКЦИЙ ВЕРОЯТНОСТНЫХ РАСПРЕДЕЛЕНИЙ

Abstract

The paper deals with the problem of clarifying the well-known inversion formulas for distribution functions, usually describing the increment of these functions. The validity of the corresponding inversion formulas for the distribution function π and their densities has been proved for the particular case of distributions with rational characteristic functions. The obtained formulas for distribution functions, which include additionally constant terms equal to 0.5, were not previously known. Functions of positively distributed random variables and quantities distributed over the entire axis have been considered separately. To test the hypothesis of fairness of the obtained treatment formula, including a previously unknown term equal to 0.5, in the general case there have been given examples of calculating distribution functions, whose characteristic functions are not considered as rational functions: for constant and uniform laws. The verification confirmed the objectiveness of the formulated hypothesis about the obtained validity of the inversion form for arbitrary distribution functions. It has also been shown that any distribution function and any density can be represented as a limit of a mixture of gamma distributions (distribution functions and densities), having shifts along the abscissa axis and, possibly, with altered signs of the arguments. The obtained result proves that the set of gamma distributions with shifted arguments is uniformly dense in the set of all distributions.