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We study a fully online matching problem with stochastic arrivals and departures. In this model, each online arrival follows a known identical and independent distribution over a fixed set of agent types. Its sojourn time is unknown in advance and follows type-specific distributions with known expectations. The goal is to maximize the weighted reward from successful matches. To solve this problem, we first propose a linear program (LP)-based algorithm whose competitive ratio is lower bounded by 0.155 under mild conditions. We further achieve better ratios in some special cases. To demonstrate the challenges of the problem, we further establish several hardness results. In particular, we show that no online algorithm can achieve a competitive ratio better than 2/3 in this model and there is no LP-based algorithm (with respect to our proposed LP) with a competitive ratio better than 1/3. Finally, we demonstrate the effectiveness and efficiency of our algorithm numerically.