This paper deals with discrete-time linear systems whose state transition is determined by a sequence of random matrices having uncertain distributions. In representing random matrices with uncertain distributions, we use random polytopes whose vertices are random matrices with given fixed distributions. For such systems characterized by random polytopes, we first tackle a problem of analyzing robust second-moment exponential stability. In particular, we show a linear-matrix-inequality-based robust stability condition that can be solved through sample-based evaluation of the associated expectations. The confidence level for such sample-based analysis is ensured by the central limit theorem. Then, we extend the results about analysis toward robust stabilization state feedback synthesis in such a way that the confidence level arguments can be facilitated. We also provide numerical examples illustrating our analysis and synthesis framework.