## 3.5 CROFTON'S METHOD FOR COMPUTING MEAN VALUESMorgan Crofton in the 1880s discovered a method for computing the mean values of certain random variables that arise in a spatial setting. Although theoretically these mean values could be computed using standard methods, occasionally Crofton's method is computationally much easier. And it is an excellent illustration of one of many special techniques devised solely for geometrical situations. The method applies to situations in which N points are
distributed Here, in our standard terminology, X is a function of N random variables (each corresponding to a location) and Crofton's method focuses on expected values of X (and functions of X) by working directly with the N points, without first deriving the probability law of X. As we will see, however, a clever application of Crofton's mean value method will allow derivation of the complete probability law for X. We illustrate Crofton's method by example. |