Application to mobile computer networks. Computer scientists often model the movement of a mobile computer as a random path within a rectangle. That is, two points are chosen at random within the rectangle, and the computer moves on a straight line from the first point to the second. In the study of mobile computer networks, it is important to know the mean length of a path (see the article “Stationary Distributions for Random Way point Models,” W. Navidi and T. Camp, IEEE Transactions on Mobile Computing, 2004:99–108). It is very difficult to compute this mean directly, but it is easy to estimate it with a simulation. If the endpoints of a path are denoted (X₁, Y₁), and (X₂, Y₂), then the length of the path is √(X₂ − X₁)² + (Y₂ − Y₁)². The mean length is estimated by generating endpoints (X^∗₁, Y^∗₁ ), and (X^∗₂, Y^∗₂) for many paths, computing the length of each, and taking the mean. This exercise presents a simulation experiment that is designed to estimate the mean distance between two points randomly chosen within a square of side 1.

a. Generate 1000 sets of endpoints (X^∗_1i, Y^∗_1i), and (X^∗_2i, Y^∗_2i). Use the uniform distribution with minimum value 0 and maximum value 1 for each coordinate of each point. The uniform distribution generates values that are equally likely to come from any part of the interval (0, 1).

b. Compute the 1000 path lengths L^∗_i = √(X^∗_2i− X^∗_1i )² + (Y^∗_2i− Y^∗_1i)².

c. Compute the sample mean path length L^→. The true mean, to six significant digits, is 0.521405. How close did you come?

d. Estimate the probability that a path is more than 1 unit long.