  Queries with aggregation and arithmetic operations, as well as incomplete
data, are common in real-world database, but we lack a good understanding of
how they should interact. On the one hand, systems based on SQL provide ad-hoc
rules for numerical nulls, on the other, theoretical research largely
concentrates on the standard notions of certain and possible answers. In the
presence of numerical attributes and aggregates, however, these answers are
often meaningless, returning either too little or too much. Our goal is to
define a principled framework for databases with numerical nulls and answering
queries with arithmetic and aggregations over them.
  Towards this goal, we assume that missing values in numerical attributes are
given by probability distributions associated with marked nulls. This yields a
model of probabilistic bag databases in which tuples are not necessarily
independent, since nulls can repeat. We provide a general compositional
framework for query answering, and then concentrate on queries that resemble
standard SQL with arithmetic and aggregation. We show that these queries are
measurable, and that their outputs have a finite representation. Moreover,
since the classical forms of answers provide little information in the
numerical setting, we look at the probability that numerical values in output
tuples belong to specific intervals. Even though their exact computation is
intractable, we show efficient approximation algorithms to compute such
probabilities.
