Properties of random numbers

A sequence of random numbers R1, R2, R3, ... used in simulation must have two essential statistical properties, uniformity and independence, plus some practical properties of a good generator.

1. Uniformity. Each number is equally likely to fall in any sub-interval of [0, 1). If the interval is divided into n equal classes, each class should receive about N/n of the N numbers. The density is f(R) = 1 for 0 ≤ R ≤ 1, so E(R) = 1/2 and Var(R) = 1/12.
2. Independence. The value of one number must not depend on any earlier or later number. There should be no pattern, trend or correlation in the sequence, so knowing R1 tells nothing about R2.
3. Maximum density. Values should cover [0, 1) closely, so the generator should have a very large number of distinct values (small gaps between possible values).
4. Long (maximum) period. The sequence should not repeat for a very long time. Ideally the period is as large as the modulus m (full period).
5. Reproducibility. The same seed must give the same sequence. This is needed for debugging and for comparing alternative designs with the same random inputs.
6. Efficiency and portability. The generator should be fast, need little memory, and give the same results on different computers.

Uniformity is checked with the chi-square or Kolmogorov-Smirnov test. Independence is checked with the runs test, autocorrelation test, gap test and poker test.