3.5 CROFTON'S METHOD FOR COMPUTING MEAN VALUES
Morgan 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 independently and uniformly in some region R of n-dimensional space. In our work, we assume that n = 1 or 2. Suppose there is some random variable X that is defined in terms of the N points and that its value depends only on the relative positions of the points, thus being invariant under translations and rotations within R. For instance, the random variable may be equal to I if no two of the points are more than a specified distance apart and zero otherwise.
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.