  One-parameter natural exponential family (NEF) plays fundamental roles in
probability and statistics. This article contains two independent results: (a)
A conjecture of Bar-Lev, Bshouty and Enis states that a polynomial with a
simple root at $0$ and a complex root with positive imaginary part is the
variance function of some NEF with mean domain $\left(0,\infty\right)$ if and
only if the real part of the complex root is not positive. This conjecture is
resolved. The positive answer to this conjecture enlarges existing family of
polynomials that are able to generate NEFs, and it helps prevent practitioners
from choosing incompatible functions as variance functions for statistical
modeling using NEFs. (b) if a random variable $\xi$ has parametric
distributions that form a infinitely divisible NEF whose induced measure is
absolutely continuous with respect to its basis measure, then there exists a
deterministic function $h$, called "reduction function", such that $\mathbb{E}
\left(h\left(\xi\right)\right)=\mathbb{V}\left(\xi\right)$, i.e.,
$h\left(\xi\right)$ is an unbiased estimator of the variance of $\xi$. The
reduction function has applications to estimating latent, low-dimensional
structures and to dimension reduction in the first and/or second moments in
high-dimensional data.
