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Re: bimodal distribution in sign of difference of Pi digits]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg51802] Re: bimodal distribution in sign of difference of Pi digits]
*From*: Roger Bagula <tftn at earthlink.net>
*Date*: Tue, 2 Nov 2004 02:06:02 -0500 (EST)
*References*: <cm21dn$gab$1@smc.vnet.net>
*Reply-to*: tftn at earthlink.net
*Sender*: owner-wri-mathgroup at wolfram.com
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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