  We consider an independently identically distributed random dynamical system
generated by finitely many, non-uniformly expanding Markov interval maps with a
finite number of branches. Assuming a topologically mixing condition and the
uniqueness of equilibrium state for the associated skew product map, we
establish a samplewise (quenched) almost-sure level-2 weighted equidistribution
of "random cycles", with respect to a natural stationary measure as the periods
of the cycles tend to infinity. This result implies an analogue of Bowen's
theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms. We
also prove another almost-sure convergence theorem, as well as an averaged
(annealed) theorem that is related to semigroup actions. We apply our results
to the random $\beta$-expansion of real numbers, and obtain almost-sure
convergences of average digital quantities in random $\beta$-expansions of
random cycles that do not follow from the application of the ergodic theorems
of Birkhoff or Kakutani. Our main results are applicable to random dynamical
systems generated by finitely many maps with common neutral fixed points.
