Guaranteed bounds for probabilistic μ

Probabilistic extensions to μ are formulated for the system context. In the application to systems these formulations are fundamentally mixed worst case and probabilistic uncertainties. A purely probabilistic μ problem is proposed and algorithms are developed and implemented to compute guaranteed upper and lower bounds. These methods are variations on branch and bound algorithms. These methods are ideally suited for the probabilistic analysis of rare events thus filling the gap between Monte Carlo methods and worst case formulations