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Traditional wireless sensor networks (WSNs) working in the license-free spectrum suffer from uncontrolled interference as the license-free spectrum becomes increasingly crowded. A WSN based on cognitive radios can potentially provide data transmissions with quality of service, while avoiding the high cost for accessing the licensed spectrum and uncontrolled interference in the license-free spectrum. Our previous works have studied delay performance of constant bit rate traffic in a cognitive radio sensor network (CRSN). In this paper we study the delay performance of random traffic with burst arrivals in the network. We derive the average delay for transmitting the random traffic using different channel sensing mechanisms, and verify the results using computer simulation. Our results show that satisfactory latency performance can be provided in the CRSN.