  In this paper we present various new inequalities for tail proabilities for
distributions that are elements of the most improtant exponential families.
These families include the Poisson distributions, the Gamma distributions, the
binomial distributions, the negative binomial distributions and the inverse
Gaussian distributions. All these exponential families have simple variance
functions and the variance functions play an important role in the exposition.
All the inequalities presented in this paper are formulated in terms of the
signed log-likelihood. The inequalities are of a qualitative nature in that
they can be formulated either in terms of stochastic domination or in terms of
an intersection property that states that a certain discrete distribution is
very close to a certain continuous distribution.
