Information on calculator Random Number Generators from PPC Journal articles

[Dieter]

(11-14-2016 01:09 AM)Namir Wrote:  Sorry, I don't have time to do a 100 run. I don't doubt you results since 5 runs is below the minimum of 30 to begin to approach normal distribution.

In the meantime I've read a few papers on testing RNGs and found an interesting quote from Donald E. Knuth's well known book "The Art of Computer Programming" where a chapter deals with random number generation and testing. I do not own this book, but he seems to suggest the following way of evaluating the Chi² value obtained after a run with, say, 10.000 of random numbers that are grouped in, say, ten categories to check their distribution in the [0;1[ domain. According to Knuth both very high and very low Chi² values are critical:

• If the Chi² value is below the 1% quantile or above the 99% quantile, the tested sample can be considered "non-random".
• If the Chi² value is between the 1% and 5% quantiles or between the 95% and 99% quantiles, the sample should be considered "suspect".
• If the Chi² value is between the 5% and 10% quantiles or between the 90% and 95% quantiles, the sample is considered "almost suspect".
  

In a way this sounds logical. Consider a die that is rolled 60 times and it comes out to exactly ten ones, ten twos etc., This would yield a Chi² value of zero and thus qualify the die as "non-random".

If a run with 10.000 random numbers is grouped into ten categories (0...0,1; 0,1...0,2; 0,2...0,3; ..., 0,9...1) the critical Chi² values for 9 degrees of freedom are:

Code:

 1%: 2,088   99%: 21,666  ("non-random")
 5%: 3,325   95%: 16,919  ("suspect")
10%: 4,168   90%: 14,684  ("almost suspect")

So let's say that for ten categories the Chi² value should fall somewhere between 4 and 15 (that's 9%/91%), or at least between 3 and 18 (3,5%/96,4%).

I did a few tests with the 9821 generator and a number of runs with 10.000 random numbers each. In most cases the Chi² value was well within the mentioned limits. Only sometimes especially the lower limit was not quite met, i.e. the numbers were a bit "too evenly distributed". The (x+pi)^3 and (x+pi)^5 RNGs were no match in this regard.

Additional tests with 20 and finally 100 categories showed essentially the same the results.

What do the math experts say?

Dieter