Using the Linear Congruential Generator (LCG) with a=67, m=31, c=17 and seed Z0 = 117generate the first FIVE random variates on [0,1]. (6 Marks)

The Linear Congruential Generator (LCG) is a type of pseudorandom number generator. It is defined by the recurrence relation:

Zn+1 = (a*Zn + c) mod m

where:

• Zn is the sequence of pseudorandom values
• a, c, and m are generator-specific constants
• "mod" is the modulus operator

In this case, a=67, c=17, m=31, and the seed Z0=117.

Let's generate the first five random variates:

1. Calculate Z1: Z1 = (aZ0 + c) mod m = (67117 + 17) mod 31 = 29
2. Calculate Z2: Z2 = (aZ1 + c) mod m = (6729 + 17) mod 31 = 14
3. Calculate Z3: Z3 = (aZ2 + c) mod m = (6714 + 17) mod 31 = 6
4. Calculate Z4: Z4 = (aZ3 + c) mod m = (676 + 17) mod 31 = 27
5. Calculate Z5: Z5 = (aZ4 + c) mod m = (6727 + 17) mod 31 = 23

Now, to convert these to the range [0,1], divide each by m:

1. Z1/m = 29/31 = 0.9355
2. Z2/m = 14/31 = 0.4516
3. Z3/m = 6/31 = 0.1935
4. Z4/m = 27/31 = 0.8710
5. Z5/m = 23/31 = 0.7419

So, the first five random variates on [0,1] are 0.9355, 0.4516, 0.1935, 0.8710, and 0.7419.