PI digit distribution - confusion

[Skogsgurra]

Can't find the proper forum for this question. Don't think there is one.

In short. We needed to explain to a customer that certain processes are stochastic rather than deterministic. He had thyristor fuses blown at very irregular intervals and it turned out that two things needed to coincide for that to happen; A. That the drive was breaking (stops in 400 milliseconds) and B. That PF compensation capacitors were switched in just then.

This happened once or twice a year and you could never tell when it was going to happen. Then, it happened just a few days apart and the customer got somewhat concerned. I thought it would be a good idea to show him that the digits of PI contain groups of digits and that those groups are at random intervals. So, we wrote a "substring finder" that searches a one million digit PI string. Our first try was the number 4711. That gave us 103 hits, which we thought was just about right.
We then tried 47111, which produced ten hits. Very much what we expected. Everything fine - so far.

Then, we wanted to show this in a diagram. Of course, we expected a uniform distribution with some variance. But what we saw was a distribution where the first hits were quite close and then spread out when we got a bit into the "one million file". Like this: ---:--:---:-----:--------:--------------:------------:--------:----- etcetera (the colons represent hits).

That was not at all what we had expected and I am glad that the customer wasn't there when we found out.

We have got the same result consistently. A few of them are given below (the numbers show where the first digit in the "search string" is positioned):

47111: 25447, 79545, 93534, 330582, 346263, 439447, 705730, 775750, 821499, 958303
47112: 73831, 299381, 375984, 718997, 962497
47113: 277993, 417209, 634628, 701464, 823702
47114: 127769, 141364, 153066, 231981, 557948, 719124, 803558, 912741, 964298, 996952
47115: 49554, 77922, 128830, 448202, 460855, 483224, 489871, 619692, 634589, 640280, 644557, 807140, 843108, 882015, 911986, 961592
47116: 36469, 92147, 273385, 279367, 300343, 318650, 378851, 483139, 546824, 635462, 685623, 782354, 803772, 872191, 888271
47117: 39880, 58449, 67914, 116125, 415961, 491234, 699587, 720026, 809578, 811719, 958795

We think (know) that the digits of PI are truly random. We think (know for sure) that our "picker" works properly. But we cannot make out why the distribution is denser at the beginning. Should it be? Why?




Gunnar Englund

http://www.gke.org

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Half full - Half empty? I don't mind. It's what in it that counts.

 

[Skogsgurra]

I'll be looking for that book! Thanks.

Gunnar Englund

http://www.gke.org

--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

 

[IRstuff]

" I was expecting that all seed numbers shall occur reasonable distributed across the one million digits"

That's what I was saying; you should not expect "reasonable" because "reasonable" is not random. Random means that things don't happen the way you expect.

TTFN
faq731-376

Need help writing a question or understanding a reply? forum1529

 

[zdas04]

Exactly. To expect any sort of distribution from a random sequence is to deny the very randomness that you are relying on. I would be much more concerned if I saw a normal distribution (or a chi squared or a student's distribution) in the data very many times.

David Simpson, PE
MuleShoe Engineering

"Belief" is the acceptance of an hypotheses in the absence of data.
"Prejudice" is having an opinion not supported by the preponderance of the data.
"Knowledge" is only found through the accumulation and analysis of data.
The plural of anecdote is not "data"

 

[SomptingGuy]

Hey, wow Gunnar! Just shone my torch into the "can't ever throw away" books and it leaped out. Not even co-written, just Sagan. Been looking for a Summer novel all Summer and (a bit late) found it. Broca's Brain was next to it too, being looking for that one for a while.

- Steve

 

[SomptingGuy]

Do not follow this link

http://kasmana.people.cofc.edu/MATHFICT/mf55-spoiler.html

- Steve

 

[Skogsgurra]

IR and David. Those two last remarks were totally unnecessary.

I was not expecting any distribution or pattern. That would have like been listening to a tune in white noise - if there is one, the noise isn't white.

But, I did see a pattern - there were too many hits in the lower portion of the one million digit string. And that is, in my view, a sign that something doesn't compute. And we found out what it was (too small a sample, even if the "sucessful" sample wasn't much larger). That is very fine with me and I do not think that I need any more explanations. My whole life has been more or less random. I know a lot about randomness.

Gunnar Englund

http://www.gke.org

--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

 

[zdas04]

Gunnar,
Of course they were unnecessary to improve your understanding of this subject, but a couple of intervening posts from other members made me think that some reinforcement of what you had already said would be helpful to the community at large. Sorry if you took my post as my thinking you needed remedial statistics.

David Simpson, PE
MuleShoe Engineering

"Belief" is the acceptance of an hypotheses in the absence of data.
"Prejudice" is having an opinion not supported by the preponderance of the data.
"Knowledge" is only found through the accumulation and analysis of data.
The plural of anecdote is not "data"