Detection of random signals under a distributed setting is considered. Due to the random nature of the spatial phenomenon being observed, the sensor decisions collected at the fusion center are correlated. Assuming that local detectors are single threshold binary quantizers, a novel approach for the fusion of correlated decisions is proposed using the theory of copulas. The proposed approach assumes only the knowledge of the marginal distribution of sensor observations but no prior knowledge of their joint distribution. Using a Neyman-Pearson (NP) framework for detection at the fusion center, the optimal fusion rule is derived. An example involving the detection of nuclear radiation is presented to illustrate the proposed approach, and results demonstrating the efficiency of the copula-based fusion rule are shown.