We consider stochastic hybrid systems (SHSs) for which the lengths of times that the system stays in each mode are independent random variables with given distributions. We propose an approach based on a set of Volterra equations to compute any statistical moment of the state of the SHS. Moreover, we provide a method to compute the Lyapunov exponents of a given degree, i.e., the exponential rate of decrease or increase at which statistical moments converge to zero or to infinity, respectively. We also discuss how, by computing the statistical moments, one can provide information about the probability distribution of the state of the SHS. The applicability of the results is illustrated in the analysis of a networked control problem with independently distributed intervals between data transmissions and delays.