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We study the problem of determining the optimal spatial node density for deployment of a Wireless Sensor Network (WSN) for distributed detection of a randomly located target in a sensing field. We formulate an optimization problem for the single cluster case and account for factors such as the Medium Access Control(MAC) protocol that is used, the wireless channel's propagation characteristics, a randomized sleep/wake-up scheduling protocol, network coverage constraints, the energy consumed, the time to reach a decision, and the total number of nodes available. We show that the optimal node density that minimizes the average Decision Error Probability (DEP) at the Cluster Head (CH) is a function of the distance from the CH and also examine trade-offs between the DEP and other network parameters as the spatial sensor density is varied across the cluster. Solution of the optimization problem and simulations are used to reveal the significant performance improvement that non-uniform spatial densities provide over uniform densities and the many optimal trade-offs that are possible amongst energy, network lifetime, detection performance and time to reach a decision.