Many of our simulations use Uniform0 1 random numbers to gen

Many of our simulations use Uniform[0, 1] random numbers to generate random values from another distribution by using the inverse probability integral transformation. One simple pseudorandom number generation method to simulate a Uniform[0, 1] random sample is the linear congruential generator. The generator is defined by the recurrence relation: Xn+1 = (a × Xn + c) (mod m) where m > 0 is the modulus, a is the multiplier, c is the increment, and X0 is the seed or starting value. Here mod is the mod function, e.g., 10 (mod 4) = 2, 9 (mod 3) = 0, 12 (mod 5) = 2. Suppose a = 67, c = 13, m = 33, and X0 = 15. Use this to generate X1, X2, …, X5, and then divide these X’s by m to get U1, U2, …, U5 as approximate uniform random numbers. What is the maximum possible number of different Ui values that this method could achieve?

Solution

Given, X0 = 15, a = 67, c = 13, m = 33.

The maximum possible number of different Ui values that this method could achieve is 32. This is because, for any number n, maximum possible value of n (mod 33) is 32.