Two streams (Rillito and Sabino Creeks) in southern Arizona were studied with particular reference to understanding their random behavior. Analysis gives unexpectedly strong evidence that certain streamflow measurements possess the statistical characteristics of a Markov chain of order no larger than six or seven. This permits statistical estimation and hypothesis testing of the parameters of the stream, leads to a simple but realistic nonlinear model of streamflow, and allows a simple method of predicting future streamflow using past streamflow measurements. In addition, statistical evidence appeared to support the contention that urbanization in the vicinity of the stream forces a trend toward dry readings at the stream's measuring station. (Modified author abstract)