Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations change according to a discrete state (jump) Markov process. The chemical master equation for such a process is typically infinite dimensional and unlikely to be computationally tractable without reduction. The recently proposed Finite State Projection technique allows for a bulk reduction of the CME while explicitly keeping track of its own approximation error. We show how a projection approach can be used to directly determine the statistical distributions for stochastic gene switch rates, escape times, trajectory periods, and trajectory bifurcations, and to evaluate how likely it is that a network will exhibit certain behaviors during certain intervals of time. We illustrate these ideas through the analysis of the switching behavior of a stochastic model of Gardner's genetic toggle switch.