Simulation and modeling - Old Questions

The linear congruential method produces a sequence of integers X₁, X₂, X₃,.......between zero and m-1 according to the following recursive relationship:

• The initial value X₀ is called the seed;
• a is called the constant multiplier;
• c is the increment
• m is the modulus

The selection of a, c, m and X₀ drastically affects the statistical properties such as mean and variance, and the cycle length.

Case 1:

     When , the form is called the mixed congruential method.

Case 2:
    When c = 0, the form is known as the multiplicative congruential method.

Case 3:

    When a = 1, the form is known as additive congruential method.

The random numbers corresponding to each random integer can be obtained as:                                        

          R_i = X_i/m, for i = 0, 1, 2, 3, …………..

Now,

Given,

    X₀ = 117, 

    α = 43,

    m=1000

We have,

For multiplicative congruential method:

        X_i+1 = (α X_i ) mod m

The sequence of random integers are calculated as follows:

X₀ = 117

X₁ = (α X₀) mod m = (43*117) mod 1000 = 5031 mod 1000 = 031

X₂ = (α X₁) mod m = (43*31) mod 1000 = 1333 mod 1000 = 333

X₃ = (α X₂) mod m = (43*333) mod 1000 = 14319 mod 1000 = 319

X₄= (α X₃ ) mod m = (43*319) mod 1000 = 13717 mod 1000 = 717