Question: Suppose two points are picked at random from the unit square. Let D be the distance between then. The main point of this problem is to find E(D). This is hard to do exactly by calculus. But some information about E(D) can be obtained as follows.
a) It is intuitively clear that E(D) must be greater than E(Dcenter), where DCenter is the distance from one point picked at random to the center of the square, and less than E(Dcorner), the expected distance of one point from a particular comer of the square. Assuming this to be the case, find the values of these bounds on E(D) using the results of Exercise 20.
b) Compute E(D2) exactly.
c) Deduce from b) a better upper bound for E(D),
d) Computer simulation of 10,000 pairs of points gave mean distance 0.5197, and mean square distance 0.3310. use these results to find an approximate 95% confidence interval for the unknowns value of E(D).