  In this paper, we consider Bayesian point estimation and predictive density
estimation in the binomial case. After presenting preliminary results on these
problems, we compare the risk functions of the Bayes estimators based on the
truncated and untruncated beta priors and obtain dominance conditions when the
probability parameter is less than or equal to a known constant. The case where
there are both a lower bound restriction and an upper bound restriction is also
treated. Then our problems are shown to be related to similar problems in the
Poisson case. Finally, numerical studies are presented.
