  We consider the class $\mathcal{E}_t(Y)$ of Appell polynomials whose
generating function is given by means of a real power $t$ of the moment
generating function of a certain random variable $Y$. For such polynomials, we
obtain explicit expressions depending on the moments of $Y$. It turns out that
various kinds of generalizations of Bernoulli and Apostol-Euler polynomials
belong to $\mathcal{E}_t(Y)$ and can be written and investigated in a unified
way. In particular, explicit expression for such polynomials can be given in
terms of suitable probabilistic generalizations of the Stirling numbers of the
second kind.
