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/8 ri  (n  1)/8.

(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  8ri.

(c) Proposal 3: Generate xi from the mixed congruential generator xn1 ≡ (5xn  7) (modulo 8), with starting value x0 4.