  This paper introduces some new characterizations of COM-Poisson random
variables. First, it extends Moran-Chatterji characterization and generalizes
Rao-Rubin characterization of Poisson distribution to COM-Poisson distribution.
Then, it defines the COM-type discrete r.v. ${X_\nu }$ of the discrete random
variable $X$. The probability mass function of ${X_\nu }$ has a link to the
R\'enyi entropy and Tsallis entropy of order $\nu $ of $X$. And then we can get
the characterization of Stam inequality for COM-type discrete version Fisher
information. By using the recurrence formula, the property that COM-Poisson
random variables ($\nu \ne 1$) is not closed under addition are obtained.
Finally, under the property of "not closed under addition" of COM-Poisson
random variables, a new characterization of Poisson distribution is found.
