In this paper we afford the fault detection and isolation problems in the context of linear discrete-time state-space models whose state equation is affected both by faults and by disturbances. Model as well as measurement errors, described as zero-mean white gaussian noises, are also assumed to additively act both on the state and on the output equations. Upon introducing several deterministic and stochastic goals, which constitute the mathematical formalization of very natural and practical requirements, necessary and sufficient conditions for the existence of an observer-based fault detector and isolator, which achieves the aforementioned goals, are finally derived.