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

Distribution of cycles for one-dimensional random dynamical systems