Numerical Characteristics of a Random Variable Related to the Engel Expansions of Real Numbers

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 p_n(x) ∈ ℕ, p_n+1(x) ≥ p_n(x) for all n ∈ ℕ. This means that p_n(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 ψ_k as a generalization of the function ψ and find their mathematical expectations Mψ_k.