There is a vast class of applications in which we know that a certain event occurred, but do not know exactly when it occurred. However, as studied by Dyreson and Snodgrass \cite{ds98}, there are many natural scenarios where probability distributions exist and quantify this uncertainty. Dekhtyar et. al. extended Dyreson and Snodgrass's work and defined an extension of the relational algebra to handle such data. The first contribution of this paper is a declarative temporal probabilistic (TP for short) calculus which we show is equivalent in expressive power to the temporal probabilistic algebra of Dekhtyar et. al. Our second major contribution is a set of equivalence and containment results for the TP-algebra. Our third contribution is the development of cost models that may be used to estimate the cost of TP-algebra operations. Our fourth contribution is an experimental evaluation of the
accuracy of our cost models and the use of the equivalence results as rewrite rules for optimizing queries by using an implementation of TP-databases on top of ODBC. (Also referenced as UMIACS-TR-2001-79)