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 with minimum value 0 and maximum value 1 for each coordinate of each point. The uniform 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.
