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Intrusion detection in a Wireless Sensor Network (WSN) is of significant importance in many applications to detect malicious or unexpected intruder(s). The intruder can be an enemy in a battlefield, or a unusual environmental change in a chemical industry etc. With uniform distribution, 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 in the deployment area. Gaussian distributed WSNs (i.e., normal distribution) can provide differentiated detection capabilities at different locations and are widely deployed in practice. In view of this, this paper analyzes the problem of intrusion detection in a Gaussian distributed WSN, by characterizing intrusion detection probability with respect to intrusion distance and network deployment parameters. Two detection models are considered: single-sensing detection and multiple-sensing detection. Effects of different network parameters on the intrusion detection probability are examined in details. This work allows us to analytically formulate the intrusion detection probability within a certain intrusion distance under various application scenarios, therefore provides insight for directing the application-specific WSN deployment such as intrusion detection.