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Performance metrics in cellular systems, such as per-user link capacity and co-channel interference, are dependent on the statistical distances between communicating nodes. An analytical model based on geometric probability in cellular systems is presented here for capacity analysis and interference estimation. We first derive the closed-form distance distribution between cellular base stations and mobile users, giving the explicit probability density functions of the distance from a base station to an arbitrary user in the same hexagonal cell, or to the users in adjacent cells. Different from numerical methods or approximation, and the existing approaches in geometric probability, this unified approach provides explicit distribution functions that can lead to all statistical moments, and is not limited by coordinate distributions, either of base stations or subscribers. Analytical results on per-user link capacity and co-channel interference are derived and validated through simulation, which shows the high accuracy and promising potentials of this approach.