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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**Re: bimodal distribution in sign of difference of Pi digits]**