Computing a family of probabilistic numbers in terms of probabilistic Stirling numbers of the second kind

In this paper, we introduce the probabilistic Bernoulli numbers, Cauchy numbers, and Euler numbers of order α associated with the random variable Y, utilizing the generating function approach. Meanwhile, by employing important tools from combinatorial analysis, such as the partial Bell polynomials and the Lagrange inversion formula, we provide computational formulas for these numbers in terms of the probabilistic Stirling numbers of the second kind. Furthermore, we introduce the probabilistic Stirling numbers of the first kind, and derive a computational formula in terms of the probabilistic Stirling numbers of the second kind, which can be seen as a probabilistic version of the Schlömilch formula.