Now that we have reviewed the basics of probabilistic modeling in Chapter 2, we are ready to focus our attention on problems of an urban nature. Many urban systems are spatially distributed and thus our analysis often must incorporate a spatial component. To accomplish this we draw upon tools of applied probability that have been developed over the years and labeled "geometrical probability" methods. However, we will see that these methods can be developed in a relatively straightforward manner from the basic principles of Chapter 2. In recalling these principles, we should keep in mind that each probabilistic modeling experiment that we confront can be approached using the following four steps:

STEP 1: Define the random variables of interest.
STEP 2: Identify the joint sample space.
STEP 3: Determine the joint probability distribution over the sample space.
STEP 4: Work within the sample space to determine the answers to any questions about the experiment.

Any special techniques that we develop, whether geometrically motivated or not, are usually only shortcuts for performing one or more of these steps. Before studying shortcuts, with their concomitant pitfalls for the novice, it is essential that one obtain a firm grounding in the basics.

As a further step toward gaining this proficiency-before launching into geometrical specialty topics-we consider a class of problems that arises in most probabilistic model building, including geometrically oriented problems. The problem involves deriving the joint probability law for one set of random variables which are expressed as functions of other random variables whose joint probability law is known. This problem, involving functions of random variables, will be our first concern in this chapter. After we have become skilled at deriving the probability laws of functions of random variables, we will (in the second part) focus our attention on spatially oriented experiments that require both knowledge of functions of random variables and special techniques associated with geometrical probability.