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In a Wireless Sensor Network (WSN), intrusion detection is of significant importance in many applications in detecting malicious or unexpected intruder(s). The intruder can be an enemy in a battlefield, or a malicious moving object in the area of interest. With uniform sensor deployment, the detection probability is the same for any point in a WSN. However, some applications may require different degrees of detection probability at different locations. For example, an intrusion detection application may need improved detection probability around important entities. Gaussian-distributed WSNs can provide differentiated detection capabilities at different locations but related work is limited. This paper analyzes the problem of intrusion detection in a Gaussian-distributed WSN by characterizing the detection probability with respect to the application requirements and the network parameters under both single-sensing detection and multiple-sensing detection scenarios. Effects of different network parameters on the detection probability are examined in detail. Furthermore, performance of Gaussian-distributed WSNs is compared with uniformly distributed WSNs. This work allows us to analytically formulate detection probability in a random WSN and provides guidelines in selecting an appropriate deployment strategy and determining critical network parameters.