  In this paper, we analyze and study a hybrid model for testing and learning
probability distributions. Here, in addition to samples, the testing algorithm
is provided with one of two different types of oracles to the unknown
distribution $D$ over $[n]$. More precisely, we define both the dual and
cumulative dual access models, in which the algorithm $A$ can both sample from
$D$ and respectively, for any $i\in[n]$,
  - query the probability mass $D(i)$ (query access); or
  - get the total mass of $\{1,\dots,i\}$, i.e. $\sum_{j=1}^i D(j)$ (cumulative
access)
  These two models, by generalizing the previously studied sampling and query
oracle models, allow us to bypass the strong lower bounds established for a
number of problems in these settings, while capturing several interesting
aspects of these problems -- and providing new insight on the limitations of
the models. Finally, we show that while the testing algorithms can be in most
cases strictly more efficient, some tasks remain hard even with this additional
power.
