Mobile edge computing has been an important supplement for traditional cloud computing architecture to offer low-delay computing services to mobile users. However, it is in general impossible for edge service providers to overdeploy so much edge resources to satisfy the rapidly increasing while diverse user demands. In this paper, we study a mobile edge computing system consisting of a service platform, cloudlets joining the system, and mobile users. In this study, we focus on a profit-driven perspective such that the service platform purchases computation resource from the resource-rich cloudlets and makes profit by processing tasks from user side. The design objective is to maximize the platform profit subject to budget constraint and stringent delay requirements for task processing. We formulate this problem as a mixed integer programming problem. Due to the NP-hardness of the problem, we design a logic based Benders decomposition algorithm as the offline solution. We further study the scenario where the task arrivals from user side and resource availability at the cloudlets are both stochastic and unknown in advance. We accordingly propose a Multi-Armed Bandit learning based resource purchasing and greedy task scheduling algorithm for the online scenario. Simulations results show the high performance of our proposed algorithms.