A discrete-time stochastic control system is considered. Here the state process depends on a control action and on a noise, which is a sequence of independent identically distributed random elements. Given the initial state and the planning horizon, sufficient conditions for the continuous dependence of the optimal reward function on the common distribution are proved for several reward criteria. This research is motivated by questions arising in problems of adaptive control of stochastic systems with unknown noise distribution, which are discussed briefly in the paper.