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Random-Number Generators - Linear Congruential Generators

5 important questions on Random-Number Generators - Linear Congruential Generators

What are the 2 objections against LCG? (LCG = Linear Congruential Generators)

1) The sequence of integers (Z_i)'s are not really random at all 2) if U_i = Z_i / m then U_i's can only take on the rationa values 0, 1/m, 2/m,....,(m-1)/m and may actually take on only a fraction of these values, while other values should occur as well

What is the recursive formula of LCG and explain the variables?

Z_i = (a*Z_(i-1) +c)*mod m where
a = multiplier c = increment m = modulus Z_0 = seed or starting value the variables are all nonnegative integers

What are the 3 conditions on m, a, c guaranteeing that pseudo-random number sequence has full period m? (thm 7.1)

1) the only positive integer that (exactly) divides both m and c is 12) if q is a prime number (divisible by only itself and 1) that divides m, then q divides a - 1
3) if 4 divides m, then 4 divides a -1
Obtaining a full or at least long period is a desriable property for a good LCG