Consider the discrete random variable X that is uniformly distributed (equal probabilities) on the set {1, 2, . . . , 9}. You wish to generate a series of random observations xi (i = 1, 2, . . .) of X. The following three proposals have been made for doing this. For each one, analyze whether it is a valid method and, if not, how it can be adjusted to become a valid method.
(a) Proposal 1: Generate uniform random numbers ri (i = 1, 2, . . .), and then set xi = n, where n is the integer satisfying n/9 < ri = (n + 1)/9.
(b) Proposal 2: Generate uniform random numbers ri (i = 1, 2, . . .), and then set xi equal to the greatest integer less than or equal to 1 + 9ri.
(c) Proposal 3: Generate xi from the mixed congruential generator xn+1 ≡ (4xn + 7) (modulo 9), with starting value x0 = 4.