posted February 10, 2007 12:23 PM            

You do not have to test all 2^32 RND's to see RND's "true"
distribution. 2^M RNDs will do (where M is any value
big enough to produce a reasonable N) as long as you also
make the number of bins in your frequency distribution 2^M.
Of course make the first M at least 5 times as big as the
second one. Thus far I have found that all Chi-squares on
such distributions to be flat zero. Note, however, that mean
RND & sd RND will not hit exactly 0.5 and sqr(1/12) by
the same device. Also, change the number of bins by
even 1 and Chi-squares will not (reliably) be 0.

You can can use the program that I posted on Feb 2, 2007 in this
thread (see page 6) to test this claim for yourself. I have
revised the program very slightly to allow very large N's, & added
a note along the lines of the first paragraph above.

NOTE added Feb 12: The results above do not hold for PBCC, but
only (for sure) for PB-DOS 3.5.

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posted February 10, 2007 01:17 PM            

There are all sorts of contradictions with the language used. If you flip a coin 100 times the “mean value” aka “expected value” for the number of heads is 50. But as others have pointed out, exactly 50 is rare. If someone said they got exactly 50, hmm ... maybe they’re stupid crooks.

Then again, the variance is the square root of 100, or 10. But again, if someone said they got exactly 90 or 110 heads, hmm ... maybe they’re smart crooks.

In 1946 back in the early days of computers RAND Corp published a book of random numbers called A Million Random Digits. They were generated by a “random frequency pulse source” so they were real world random, not computed pseudo-random.

I read – somewhere I can’t remember – when the engineers did the first run, they were surprised that the distribution wasn’t more or less nearly uniform, so they cooked up numbers to even it out. Probably later editions are more sophisticated and leave the numbers alone.

The problem is: one knows that in the book there are more even than odd numbers (or vice versa). Don’t give away the plot to someone else.

This points up the futility of ... of what? Rational analysis of probability? The fact that there is no valid concept of the probability of something?

Tatyana Ehrenfest-Afanasyeva (Russian, she later lived in Germany then Holland) wrote about this in “On the Use of the Notion ‘Probability’ in Physics,” American Journal of Physics 26:388 (1958) and you can read it here: www.physics.ucla.edu/~cwp/articles/Ehrenfest.html

Some amusement can be had by reading the reviews of the latest, 2002 edition, of A Million Random Digits at Amazon.com: www.amazon.com/Million-Random-Digits-Normal-Deviates/dp/0833030477

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