Many of our simulations use Uniform[0, 1] random numbers to generate random valu

Question

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?

Explanation / Answer

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.

i Xi = a x Xi-1 + c Ui = Xi (mod 33) 0 15 15 1 1018 28 2 68219 8 3 4570686 21 4 306235975 1 X5 20517810338 14

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