  We investigate the brightness distribution expected for thermonuclear
explosions that might result from the ignition of a detonation during the
violent merger of white dwarf (WD) binaries. Determining their brightness
distribution is critical for evaluating whether such an explosion model could
be responsible for a significant fraction of the observed population of SNe Ia.
We argue that the brightness of an explosion realized via the violent merger
model is determined by the mass of nickel 56 produced in the detonation of the
primary WD. We use a set of sub-Chandrasekhar mass WD detonation models to
derive a relationship between primary WD mass and expected peak bolometric
brightness. We use this relationship to convert the masses of merging primary
WDs from binary population models to a predicted distribution of explosion
brightness. We find a striking similarity between the shape of our theoretical
peak-magnitude distribution and that observed for SNe Ia: our model produces a
brightness distribution that roughly covers the range and matches the shape of
the one observed for SNe Ia. However, this agreement hinges on a particular
phase of mass accretion during binary evolution: the primary WD gains
~0.15-0.35 Msun from a slightly-evolved helium star companion. We also find
that with high probability, violent WD mergers involving the most massive
primaries (which should produce bright SNe) have delay times <500 Myr
(Abridged).
