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

*To*: mathgroup at smc.vnet.net*Subject*: [mg51833] Re: bimodal distribution in sign of difference of Pi digits]*From*: Roger Bagula <tftn at earthlink.net>*Date*: Wed, 3 Nov 2004 01:24:26 -0500 (EST)*References*: <cm21dn$gab$1@smc.vnet.net> <cm7cv1$lh6$1@smc.vnet.net>*Reply-to*: tftn at earthlink.net*Sender*: owner-wri-mathgroup at wolfram.com

The correct probability is {40/90,10/19,40/90) for the Sign types (1,0,-1). I made a subtraction mistake. The model of the probabilities becomes: ( with no zero state ) f[n_]:=f[n]=f[n-1]+Random[Integer,{1,40}]/90-Random[Integer,{1,40}]/90 f[0]=Random[Integer,{1,40}]/90-Random[Integer,{1,40}]/90 My friend has used stepwise calculation in Mathematica to go to a very high number of Pi digits ( 10 of millions) and the deviations from zero still remains and grows. It appears there is no "ideal" of randomness that can be reached by our current methods of calculation. 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