  Liverani-Saussol-Vaienti (L-S-V) maps form a family of piecewise
differentiable dynamical systems on $[0,1]$ depending on one parameter
$\omega\in\mathbb R^+$. These maps are everywhere expanding apart from a
neutral fixed point. It is well known that depending on the amount of expansion
close to the neutral point, they have either an absolutely continuous invariant
probability measure and polynomial decay of correlations ($\omega <1$), or a
unique physical measure that is singular and concentrated at the neutral point
($\omega >1$). In this paper, we study the composition of L-S-V maps whose
parameters are randomly sampled from a range in $\mathbb R^+$, and where these
two contrasting behaviours are mixed. We show that if the parameters $\omega<1$
are sampled with positive probability, then the stationary measure of the
random system is absolutely continuous; the annealed decay rate of correlations
is close (or in some cases equal) to the fastest rate of decay among those of
the sampled systems; and suitably rescaled Birkhoff averages converge to limit
laws. In contrast to previous studies where $\omega \in [0,1]$, we allow $
\omega >1$ in our sampling distribution. We also show that one can obtain
similar decay of correlation rates for $\omega \in [0,\infty)$, when sampling
is done with respect to a family of smooth, heavy-tailed distributions.
