This paper discusses the H- index problem for stochastic linear discrete-time systems. A necessary and sufficient condition of H- index is given for such systems in finite horizon. It is proved that when the H- index is greater than a given value, the feasibility of H- index is equivalent to the solvability of a constrained difference equation. The above result can be applied to the fault detection observer design. Finally, some examples are presented to illustrate the proposed theoretical results.