It is known that any number x ∈ (0; 1] ≡ Ω has a unique Engel expansion $$ x=\sum \limits_{n=1}^{\infty}\frac{1}{\left({p}_1(x)+1\right)\dots \left({p}_n(x)+1\right)}, $$ where pn(x) ∈ ℕ, pn+1(x) ≥ pn(x) for all n ∈ ℕ. This means that pn(x) is a well-defined measurable function on the probability space (Ω, ℱ, λ), where ℱ is the σ-algebra of Lebesgue-measurable subsets of Ω and λ is the Lebesgue measure. The main subject of our research is a function $$ \psi (x)=\sum \limits_{n=1}^{\infty}\frac{1}{p_n(x)+1}, $$ defined on Ω* ⊂ Ω, where Ω* is the set of convergence of the series $$ {\sum}_{n=1}^{\infty}\frac{1}{p_n(x)+1} $$ . It is proved that the function ψ is defined (takes finite values) a.e. on (0; 1]. Moreover, it is shown that ψ is a random variable on the probability space (Ω*, ℱ∗, λ), where ℱ∗ is the σ-algebra of Lebesgue-measurable subsets of Ω*. We determine the mathematical expectation and variance of the function ψ. Furthermore, we consider random variables ψk as a generalization of the function ψ and find their mathematical expectations Mψk.

中文翻译：

---

与实数的恩格尔展开相关的随机变量的数值特征

其中 ℱ∗ 是 Ω* 的勒贝格可测子集的 σ-代数。我们确定函数 ψ 的数学期望和方差。此外，我们将随机变量 ψk 视为函数 ψ 的推广，并找到它们的数学期望 Mψk。