  The paper studies distributed binary hypothesis testing over a two-hop relay
network where both the relay and the receiver decide on the hypothesis. Both
communication links are subject to expected rate constraints, which differs
from the classical assumption of maximum rate constraints. We exactly
characterize the set of type-II error exponent pairs at the relay and the
receiver when both type-I error probabilities are constrained by the same value
$\epsilon>0$. No tradeoff is observed between the two exponents, i.e., one can
simultaneously attain maximum type-II error exponents both at the relay and at
the receiver. For $\epsilon_1 \neq \epsilon_2$, we present an achievable
exponents region, which we obtain with a scheme that applies different versions
of a basic two-hop scheme that is optimal under maximum rate constraints. We
use the basic two-hop scheme with two choices of parameters and rates,
depending on the transmitter's observed sequence. For $\epsilon_1=\epsilon_2$,
a single choice is shown to be sufficient. Numerical simulations indicate that
extending to three or more parameter choices is never beneficial.
