Stochastic models have been proposed as one technique for generating scenarios of future climate change. One particular daily stochastic weather generator, termed Richardson's Model or WGEN, has received much attention. Because it is expressed in a conditional form convenient for simulation (e.g., temperature is modeled conditional on precipitation occurrence), some of its statistical characteristics are unclear. In the present paper, the theoretical statistical properties of a simplified version of Richardson's model are derived. These results establish that when its parameters are varied, certain unanticipated effects can be produced. For instance, modifying the probability of daily precipitation occurrence not only changes the mean of daily temperature, but its variance and autocorrelation as well. A prescription for how best to adjust these model parameters to obtain the desired climate changes is provided. Such precautions apply to conditional stochastic models more generally.