In event-driven wireless sensor networks, the network lifetime has a random nature due to the randomness of data reporting. The lifetime is even more nondeterministic when sensors are also deployed randomly. The lifetime of such a network is influenced by node deployment, initial energy of sensors, packet generation model and the number of sensors. This work quantifies the effect of these parameters on the lifetime of randomly deployed event-driven networks. First, the lifetime of individual sensors are studied. Then, an analytical expression is obtained for the complementary cumulative density function of the network lifetime. Such an analysis can be used for choosing the network parameter and efficiently optimizing the network lifetime. The results of this work are obtained for both multi- hop and single-hop wireless sensor networks and are verified with computer simulation. The approaches of this paper are shown to be applicable to more general cases.