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The digits of pi are not random

10 点作者 seccode8 个月前

5 条评论

eesmith8 个月前
Next try 10 million digits instead of 10 thousand.<p>The Z-score cannot be interpreted so easily. It must be evaluated over all the different ways people have tried to test the digits for randomness.<p>&quot;Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.&quot; - <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Pi" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Pi</a>
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lifthrasiir8 个月前
Z-score is not a direct measure of statistical significance, which necessarily needs a correct interpretation of z-score or similar numbers. If my measurement of normally distributed random variable resulted in the z-score of 5.0, it can still occur by a slim (&lt;0.000058%) but still pure chance. If we don&#x27;t know the variable&#x27;s distribution in advance, the chance can be as high as 4% per Chebyshev&#x27;s inequality. You don&#x27;t have any necessary interpretation to derive your conclusion to exclude such possibilities.<p>The only thing one can conclude from your claim is that, the first handful number of digits in pi can be very, very slightly compressed if you are somehow able to ignore the size of that classifier [1]. The algorithmic randomness however requires the classifier to be included, and there is no known self-contained program that is smaller than printing all digits in verbatim or computing them in the first place.<p>[1] By the way, this is not really surprising at all. In fact there are 5,001 (<i>not</i> 5,000) out of first 10,000 decimal digits of pi that are between 5 and 9, so it can be very, very slightly compressed in that way---of course after the decompressor excluded.
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tgv8 个月前
One obvious problem is the model. There is a model that can predict each digit with 100% accuracy, so you must show in advance that your model will never have a bias, in <i>all instances</i>, not a single run. Even then, the tiniest error in the randomization procedure can have an effect. Perhaps you&#x27;ve only shown that your particular implementation of random forests is biased.<p>The second problem is the interpretation of statistical significance. You don&#x27;t even explain how you compute it, and it&#x27;s not in the code. For starters, what&#x27;s your df?<p>Another problem is the sample. It might be possible that any range of 10,000 digits isn&#x27;t random (according to your criterion), but the whole is. Now that I think about it: it is very likely there is a structure, since it&#x27;s a rather limited string, and the number of possible bit strings of length n quickly outnumbers the total length of the string for increasing n. 10k bits can only contain the unique strings up to length 10, and those wouldn&#x27;t be random at all (since you can predict the remaining bits with increasing accuracy). So the details of your prediction model are really essential.
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seccode8 个月前
Worth noting that I showed _statistical significance_, not _proof_
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jqpabc1238 个月前
Pi = circumference &#x2F; diameter<p>Pi is irrational but definitely not random.