Simulation and Modelling - Old Questions

9. Use Multiplicative congruential method to generate a sequence of 10 three-digit random integers and corresponding random variables. Let X₀ = 5, a = 3 and c=2. 

Given,

    X₀ = 5, 

    α = 3

    c = 2,

    For three-digit random integers

        m=1000

We have,

For multiplicative congruential method ( c =0) so

        X_i+1 = (α X_i ) mod m

The sequence of random integers are calculated as follows:

X₀ = 5

R₀ = 5/1000 = 0.005

X₁ = (α X₀) mod m = (3*5) mod 1000 = 15 mod 1000 = 15

R₁ = 15/1000 = 0.015

X₂ = (α X₁) mod m = (3*15) mod 1000 = 45 mod 1000 = 45

R₂ = 45/1000 = 0.045

X₃ = (α X₂) mod m = (3*45) mod 1000 = 135 mod 1000 = 135

R₃ = 135/1000 = 0.135

X₄= (α X₃ ) mod m = (3*135) mod 1000 = 405 mod 1000 = 405

R₄ = 405/1000 = 0.405

X₅= (α X₄ ) mod m = (3*405) mod 1000 = 1215 mod 1000 = 215

R₅= 215/1000 = 0.215

X₆= (α X₅) mod m = (3*215) mod 1000 = 645 mod 1000 = 645

R₆ = 645/1000 = 0.645

X₇= (α X₆) mod m = (3*645) mod 1000 = 1935 mod 1000 = 935

R₇ = 935/1000 = 0.935

X₈= (α X₇) mod m = (3*935) mod 1000 = 2805 mod 1000 = 805

R₈ = 805/1000 = 0.805

X₉= (α X₈) mod m = (3*805) mod 1000 = 2415 mod 1000 = 415

R₉ = 415/1000 = 0.415

X₁₀ =  (α X₉) mod m = (3*415) mod 1000 = 1245 mod 1000 = 245

R₁₀ = 245/1000 = 0.245