Statistics of spatial average in a random two-state medium

A one-dimensional medium consisting of two alternating states is first studied. The occurrences of the states as well as the respective durations are assumed random according to a renewal process. The occurrence model is then extended to a two-dimensional medium by using appropriate two-dimensional variance function for the discrete random field. The properties associated with the two states could be drastically different; they are also subject to uncertainties and spatial variation. A random variable or a random field could be used to model each material property. Satisfactory performance of a system associated with such medium could depend on the spatial averageover some domain within the medium, whose statistics are derived. These statistics would serve as inputs to reliability evaluation including the determination of exceedance probability of a given threshold by some local spatial average property. The effect of the shape of the rectangular averaging domain is studied by varying its aspect ratio. Results could be readily extrapolated to a three-dimensional medium by modifying the corresponding variance functions. Examples of two-state systems in engineering are cited that could apply the results of the proposed model. It could also be used to study the effect on the performance reliability due to possible presence of an anomalous state embedded within an otherwise statistically homogenous medium.