Here is a probabilistic method for computing the area of a given subset S of the unit square. The method uses a sequence of independent random selections of points in the unit square [0,1]•[0,1], according to a uniform probability law. If the ith point belongs to the subset S the value of a random variable Xi is set to 1, and otherwise it is set to 0. Let X1,X2,... be the sequence of random variables this defined, and for any n, let Sn=(X1+X2+...+Xn)/n. a)Show that E[Sn] is equal to the area of the subset S, and that var(Sn) diminishes to 0 as n increases. b)Show that to calculate Sn, it is sufficient to know Sn-1 and Xn, so the past values of Xk, k=1,...,n-1, do not to be remembered. Give a formula. c) Write a program to generate Sn for n=1,2,...,10000, using the computer’s random number generator, for the case where the subset S is the circle inscribed within the unit square. How can you use your program to measure experimentally the value of pi. d) Use a similar computer program to calculate approximately the area of the set of all (x, y) that lie within the unit square and satisfy 0<=cos(pi•x) + sin(pi•y) <= 1