It is instructive to review some simplified cases which can assist us in leading up to the solution of the multidimensional problem. Example 1 (Laplace’s problem): Two people agree to meet at a restaurant between 2:00 and 3:00. They can arrive at some random time between the set times and agree, if either should arrive first, to wait 10 minutes for the other party. We now wish to calculate the probability of the meeting taking place. Although the problem appears one-dimensional, it is important to realize that this problem is intrinsically three dimensional. For the first person, the arrival time lies along the x axis with unit probability density function measured along the z axis. A similar situation exists for the second person (arrival time lying along the y axis). Thus, the calculated probability is a “volume”* bounded by the joint probability density function (represented as a surface along the z axis) and the “rules of meeting” limits in the x-y plane. The role of the z axis in this problem is often ignored.* Referring to Figure 2, the problems solves by assigning the time between 2:00 and 3:00 to a unit square: It must be remembered that the volume integration is performed in the Lebesque sense
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