Occupancy Numbers in Testing Random Number Generators
The classical occupancy problem where n balls are placed in N cells is used for testing of random number generators. We show that the statistics of appropriately chosen occupancy numbers are incompatible with the statistics of manypseudorandom number generators (PRNGs) even if they are truncated. More than that, the incompatibility shows up on relatively small samples long before the period of the PRNG is reached. We introduce generalized Fermi--Dirac models as idealized models for PRNGs. These models are used to make educated guesses of how large the sample sizes should be to detect the deficiencies of the PRNGs under study. We use the developed occupancy tests together with some other ones to examine the performance of several widely used random number generators, including random of the UNIX C library. We found that random failed two of the conducted tests rather badly. We also tested a true random number generator based on $\alpha$-decay which passed all our relevant tests successfully.