  Let Y be a random variable whose degenerate moment generating functions exist
in some neighborhoods of the origin. The aim of this paper is to study the
probabilistic degenerate Stirling numbers of the first kind associated with Y
which are constructed from the degenerate cumulant generating function of Y.
They are a degenerate version of the probabilistic Stirling numbers of the
first kind associated with Y, which were recently introduced by Adell-Benyi. We
investigate some properties, related identities, recurrence relations and
explicit expressions for those numbers. In addition, we apply our results to
the special cases of normal and gamma random variables.
