Fog radio access network (Fog-RAN), which pushes the caching and computing capabilities to the network edge, is capable of efficiently delivering contents to users by using carefully designed caching placement and content replacement algorithms. In this paper, the transmission scheme design and coding parameter optimization will be considered for coded caching in Fog-RAN, where the reliability of content delivery, i.e., content outage probability, is used as the performance metric. The problem will be formulated as a complicated multi-objective probabilistic combinatorial optimization. A novel maximum b-matching approach will then be proposed to obtain the Pareto optimal solution with fairness constraint. Based on the fast message passing approach, a distributed algorithm with a low memory usage of O(M + N) is also proposed, where M is the number of users and N is the number of Fog-APs. Although it is usually very difficult to derive the closed-form formulas for the optimal solution, the approximation formulas of the content outage probability will also be obtained as a function of coding parameters. The asymptotic optimal coding parameters can then be obtained by defining and deriving the outage exponent region (OER) and diversity-multiplexing region (DMR). Simulation results will illustrate the accuracy of the theoretical derivations, and verify the outage performance of the proposed approach. Therefore, this paper not only proposes a practical distributed Fog-AP selection algorithm for coded caching, but also provides a systematic way to evaluate and optimize the performance of Fog-RANs.