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Re: bimodal distribution in sign of difference of Pi digits]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg51845] Re: bimodal distribution in sign of difference of Pi digits]
*From*: Roger Bagula <tftn at earthlink.net>
*Date*: Wed, 3 Nov 2004 01:25:36 -0500 (EST)
*Organization*: tftn/bmftg
*References*: <cm21dn$gab$1@smc.vnet.net> <cm7cv1$lh6$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
http://news.bbc.co.uk/1/hi/sci/tech/2146295.stm
http://www.pnl.gov/energyscience/10-01/art3.htm
http://pi.nersc.gov/
I've just trying to answer the question raised by Dr. David Bailey?
My answer is a qualified No to his question about Pi digits being random.
But they are the best "nature" non-Markovian Rabndomness we have.
As his ( Bailey's) formula gives individual digits using a PSLQ
derivation in base 16,
It is no they aren't techically dependent base 16 on the previous digits.
There are a lot of transcendental numbers and their technology is only
just being explored.
The estimate is that the number of transcentals is very like that of the
rational numbers,
but we don't have a Farey tree and it's sequences like in
transcendenatls yet.
The point is that Pi is a "benchmark" for randomness.
CA 30 just isn't good enough to be.
And yes there is a dependence in Pi digits, but it isn't considered
algebraic
even though PSLQ is used to compute it:
http://library.wolfram.com/infocenter/MathSource/4263/
http://mathforum.org/epigone/comp.soft-sys.math.mathematica/jolpoxhen
http://www.lbl.gov/Science-Articles/Archive/pi-algorithm.html
It can't be algebriac or Pi wouldn't be transcendental.
Roger Bagula wrote:
>I'm learning to analyze this type of problem.
>The sum comes down to a cumlative integer probality sum ( Sign is only
>integer).
>I simulated it using the (a,b) two simple probabilities of 10 symbols to
>get
>(7/18,1/9,7/18) out 90 possible states.( 2*Binomial[10,2])
>The result behaves just as the other digits simulations did without
>using the digits:
>I also realize that "independent" probabilities may be an "ideal" myth
>as nothing comes from nowhere, but still is is the ideal from such
>probaility as a thought experiment.
>Thus, using a pseudorandom that is in any tinture Markov or dependent
>on it's history is
>a "fault" to the simulation.
>The trouble is we actually lack an ideal probability type pseudorandom.
>No such algorithm exist as far as I know
> or have been able to search out in the last 30 years of study.
>(* simulation of 10's digits equal probabuility (a,b) independently*)
>(* using (7/18,1/9,7/18) as probabilities that the Sign of the
>difference is (1,0,-1)*)
>digits=50000
>SeedRandom[Random[Integer,digits]]
>f[n_]:=f[n]=f[n-1]+Random[Integer,{0,7}]/18-Random[Integer,{0,7}]/18
>f[0]=Random[Integer,{0,7}]/18-Random[Integer,{0,7}]/18
>a=Table[Floor[f[n]],{n,1,digits}];
>ListPlot[a,PlotJoined->True]
>b=Flatten@{0,Length/@Split[Sort@a], 0}
>ListPlot[b,PlotJoined->True];
>
>Roger Bagula wrote:
>
>
>
>>Dear jasonp,
>>I don't know.
>>This method is a new way to investigate Pi digits.
>>I had done some counts of base ten digits frequencies before this.
>>I have no real explaination of why the difference is higher in higher number of digits.
>>The groups of positive "Sign"s should
>>random. It is Sign[x]-> {-1,0,1} depending on the difference in consecutive
>>differences. It is the probability of a digit pair:
>>{a,b}--> Sign[a-b]
>>p=Probability [a]*Probability[b]
>>If they are equal as p0:
>>p->p0^2
>>If the Mathematica for such a probability would be:
>>p0->Random[Integer,{0,9}] as a Distribution
>>Since this is an straight type probabilty and not a Gaussian
>>the probabilies are equal and should be over a long term
>>1/10 each or a total of
>>p-->1/100
>>different for different combinations:
>>{a>b}->+1,{a-1},{a=b}->0
>>at {4/10,4/10,2/10} that gives something like
>>4/1000,4/1000,2/1000
>>I'm not seeing that kind of behavior except for the bimodal
>>which is expected as
>>(a=b) is
>>only about 2/10 of the 1/100 and I'm seeing more zeros than that.
>>It appears to be a much more complex distribution.
>>I want to try E and orther irrational numbers by this method as well!
>>I'm glad you asked as I hadn't thought to do a probability analysis
>>until now!
>>I can simulate the probability above in Mathematica
>>and see what I get
>>and compare them.
>>jasonp at boo.net wrote:
>>
>>
>>
>>
>>
>>>Quoting Roger Bagula :
>>>
>>>
>>>
>>>
>>>
>>>>(* Sum of the sign of the differences between the first 2000 digits of Pi*)
>>>> >>
>>>>
>>>>
>>>Shouldn't this behave like a random walk, i.e. the variance
>>>increases over time?
>>>
>>>jasonp
>>>
>>>
>>>------------------------------------------------------
>>>This message was sent using BOO.net's Webmail.
>>>http://www.boo.net/
>>>
>>>
>>>
>>>
>>>
>>
>>
>
>
>
--
Respectfully, Roger L. Bagula
tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn at netscape.net
URL : http://home.earthlink.net/~tftn
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