  Carlitz [2] initiated a study on degenerate versions of Bernoulli and Euler
numbers which has been extended recently to the researches on various
degenerate versions of quite a few special numbers and polynomials. They have
been explored by using several different tools including generating functions,
combinatorial methods, $p$-adic analysis, umbral calculus, special functions,
differential equations and probability theory as well. \par The degenerate
Poisson random variables are degenerate versions of the Poisson random
variables. In [6], studied are the degenerate binomial and degenerate Poisson
random variables in relation to the degenerate Lah-Bell polynomials. Among
other things, it is shown that the rising factorial moments of the degenerate
Poisson random variable are expressed by the degenerate Lah-Bell polynomials.
Also, it is shown that the probability-generating function of the degenerate
Poisson random variable is equal to the generating function of the degenerate
Lah-Bell polynomials. The zero-truncated Poisson distributions (also called the
conditional or the positive Poisson distributions) are certain discrete
probability distributions whose supports are the set of positive integers. In
[10], the zero-truncated degenerate Poisson random variables, whose probability
mass functions are a natural extension of the zero-truncated Poisson random
variables, are introduced and various properties of those random variables are
investigated. Specifically, for those distributions, studied are its
expectation, its variance, its n-th moment, its cumulative distribution
function and certain expressions for the probability function of a finite sum
of independent degenerate zero-truncated Poisson random variables with equal
and unequal parameters.
