We develop a new class of event detection algorithms in Wireless Sensor Networks where the sensors are randomly deployed spatially. We formulate the detection problem as a binary hypothesis testing problem and design the optimal decision rules for two scenarios, namely the Poisson Point Process and Binomial Point Process random deployments. To calculate the intractable marginal likelihood density, we develop three types of series expansion methods which are based on an Askey-orthogonal

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... ls. In addition, we develop a novel framework to provide guidance on which series expansion is most suitable (i.e., most accurate) to use for different system parameters. Extensive Monte Carlo simulations are carried out to illustrate the benefits of this framework as well as the quality of the series expansion methods, and the impacts that different parameters have on detection performance via the Receiver Operating Curves (ROC).