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We analyze the quality of monitoring (QoM) of stochastic events by a periodic sensor which monitors a point of interest (PoI) for q time every p time. We show how the amount of information captured at a PoI is affected by the proportion q/p, the time interval p over which the proportion is achieved, the event type in terms of its stochastic arrival dynamics and staying times and the utility function. The periodic PoI sensor schedule happens in two broad contexts. In the case of static sensors, a sensor monitoring a PoI may be periodically turned off to conserve energy, thereby extending the lifetime of the monitoring until the sensor can be recharged or replaced. In the case of mobile sensors, a sensor may move between the PoIs in a repeating visit schedule. In this case, the PoIs may vary in importance, and the scheduling objective is to distribute the sensor's coverage time in proportion to the importance levels of the PoIs. Based on our QoM analysis, we optimize a class of periodic mobile coverage schedules that can achieve such proportional sharing while maximizing the QoM of the total system.