Binomial distribution in VMS

[bi3mw]

In contrast to the herbal section the substitution of the letter groups, applied to the entire VMS, improves the binomial distribution.

Distribution without substitution:
   

Distribution with substitution:

ch -> C
sh -> S
cth -> T
ckh -> K
cph -> P
cfh -> F


   

[bi3mw]

An addition to @RobGea`s post #44:

Total occurrences of the specified substrings: 17722
Total number of words in the text: 39020
Ratio of total occurrences to total words: 45.42%

Example of words (or parts of words) containing substrings more than once:
chckhy: 149
chcthy: 89
shckhy: 62
checkhy: 52
sheckhy: 38
shcthy: 33
checthy: 31
chckhey: 30
chockhy: 26
chocthy: 25
shecthy: 21
chokchy: 21
chotchy: 14
chckhdy: 13
shocthy: 13
............

Substring combinations (sorted by frequency):

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ch, ckh: 385
Words: chckhol, ckhochy, chckhy, chckhy, oechockhy, chockhy, chckhy, chckhan, chckhy, chckhy, chckhy, chckhy, dchckhy, chckhhy, chckhy, chckho, chockhol, chckhey, chckhy, chckhol, chckhy, chockhy, chckhaly, dchckhy, chckhy, checkhy, checkhey, checkhy, chckhol, chckhy, chckhy, chckhy, chckhy, chckhy, checkhy, olchckhy, chckhy, chckhdy, chckhy, checkhy, chckhor, chockhy, chckh, chckh, checkhy, chckhdy, chckhy, chckhy, chckhy, chckhd, chckhy, chckhy, chckhy, chckhy, chckheody, chockhy, chckhy, checkhy, chckhhhy, checkhy, chckhy, dchckhy, chckhy, chckhy, chckhy, chckhy, chckham, chockhdy, chckhy, chckhdy, cheockhy, chckhedy, chckhy, chckhy, chckhy, checkhy, chckhey, chckhy, chckhy, chckhey, ochockhy, chckheey, chckhey, chckhy, chckhy, chckhy, chckhy, chckhdy, cheeckhody, chockhy, chckhey, dcheckhey, chockhy, chockhey, cheockhey, ycheockhy, chckhy, chckhhy, chckhedy, chckhy, chckhd, chckhy, chockhy, cheockhey, chockhey, chockhy, chckhy, chockhy, dycheckhy, cholckhhy, chckhhy, chckhy, chekackhy, cheockhy, chockhy, chckhy, checkhy, chckhy, chckhy, chckhy, chckhy, chckhy, chckhey, chckhy, chckhey, chckheed, checkhy, chckh, chckhy, chckhy, chckhy, chckhy, chckhy, chckhy, chckhy, checkhey, chckhey, checkhey, checkhy, checkhy, checkhy, chckhy, checkhy, chckhy, chckhedy, chckhey, chckhy, checkhy, checkhy, chckhy, chckhy, checkhy, chckhd, chckhedy, checkhy, chckhdy, dchckhedy, chckhdy, chckhal, chckhy, checkhy, lcheckhy, lcheckhey, chckhey, chckhey, checkhy, chckhes, chckhy, ycheckhey, chckhy, checkhy, chckhey, chckhy, checkhy, qokechckhy, chckhy, chckhy, checkhy, checkhy, chckhy, chckhy, chckhy, chckhy, chckhy, chckhy, checkhy, chckhy, lcheckhy, lcheckhedy, chckhy, checkho, chckhey, lcheckhy, chckhy, chckhdy, checkhy, checkhy, chckhy, dalchckhy, checkhy, chckhdy, chckhal, chckhy, checkhy, checkhdy, chckhedy, chckhdy, schckhy, chckhy, chckhy, chckhy, checkhy, chckhy, chckhy, chckhedy, chckhedy, chckhor, chckhy, ychckhy, chckhdy, chckhdy, chckhey, chckhy, chckhy, chckhy, cheockhy, chckhy, checkhy, chckhey, dcheeckhos, cheockhy, chckhy, chockhy, cheockhy, checkhy, cheockhy, chockhy, chckhhhy, dolchsyckheol, chckhy, chockh, chockhy, chockhor, chckhs, ychockhy, chckhaiin, opchackhy, chckhy, chckhy, chckhy, chckhy, chckhy, chckhey, chckhy, chockhy, checkhey, chockhia, checkhey, chockhy, cheockhy, chckheody, chockhey, chockhy, chockhhor, chckhey, chockhar, checkhey, cheeckhey, chockhy, chckhy, chockhey, chckhy, checkhy, chckhy, chckhy, chckhey, chckhy, checkhy, chckhy, chckhey, checkhy, chckhy, checkhy, chckhey, chckhy, chckhy, chockhar, chckhey, chckhey, chckhy, checkhd, chockhy, chckhdy, chockhedy, chckhodu, chckhdy, checkhy, chckhy, cheockhy, chckhy, chckhy, cheockhy, chckhy, chckhy, chockhed, chckhedy, chckhy, chckhy, chckhom, chockhey, checkhy, chckhy, chckhy, checkhy, checkhed, checkhey, chckhey, checkhedy, checkhy, chckhy, chckhhy, checkhey, chckhey, chckhy, chckhy, chckhedy, checkhed, chckhy, chckhy, chckhal, chckhy, chckhaiin, chckheal, checkhy, checkhy, chckhy, chckhedy, checkhey, chckhy, chckhal, chckhy, chckhey, chckhy, chckhy, chckhy, checkhy, checkhey, checkhy, checkhy, checkhol, checkhy, chckhy, checkhey, chckhhy, chckhody, ycheeckhy, chckhy, chckhey, chckhey, chockhy, pcheockhy, chckhhy, checkhy, chockhy, checkhol, chckhedy, chckhhy, chckhaiin, chckhey, chckhd, chockhy, chckhy, chockhy, cheockhy, chckhey, chockhy, lchckhy, chckhy, chckhy, checkhy, chckhol, chckhy, chckhy, chckhey, chckhy

ch, ch: 322
Words: chotchey, chotchey, chopchy, chokchy, chopchol, chapchy, chekchoy, charochy, chotchy, chochy, chochy, choschy, chorcheor, chokchy, kchekchy, rchealcham, chykchy, chokchy, scharchy, chchotaiin, chokcho, chodykchy, qopchypcho, chorchy, chopchal, chtchor, ochkchor, dchokchy, chokchy, ychykchy, chpchol, chpchey, chokchy, chypchy, chkchy, ychekchy, dchchy, chypcham, cholchom, chykchy, chokchy, ychekch, chkchy, chpcheos, chodchy, pchofychy, chofchy, chotchy, kchochaiin, choldchy, dchchy, chkchy, chotchy, chypchey, chedyfchy, dchokchy, chodchy, chotchy, dchschy, chschsy, chotchy, chotchy, otcholchy, chchoty, chetchy, chokchaiin, chkchean, chtchey, chepchy, chotchol, chakchy, chokcheey, chefchol, chetchey, tcheychedy, chofchody, chokchy, chechdaiin, pcheoepchy, chefalchhs, cholchedy, chefchy, chdalchdy, chechey, chochy, chetchaiin, dchokchy, chkcheeu, chotchy, dchytchy, ykchotchy, pchotchy, kchochor, orchochor, chetchy, chochor, ychdykchdy, chdchy, chepchefy, chkchory, chokchy, chtchy, chkchy, cheotchar, chekchey, chotcheey, chkchy, choetchy, cheolkeepchy, chekcheey, wcheekchy, tchtcho, chepchx, chokchol, chtchy, chopchy, chkchy, chotcho, chochy, pcheodchy, chotchy, chotcheol, chotchy, chokchol, chochy, chotchy, chokchy, chotchy, fchochor, chotchy, chokchodaiin, chokchy, tchotchey, chotchody, cheokchet, chokchy, chetcho, chkalchy, chefchdy, chochar, pchtchdy, kcheedchdy, chchswy, chorchy, chokchey, chokchol, chochor, chotchey, chokcheo, chokchy, chokchol, kchokchy, chokchy, chokcho, chotchey, pchchy, chofchol, chepchey, chefchy, cholpchd, chekchy, chkchody, dchschdy, chekchy, chkchdar, chykchs, dchesykchy, cheokorchey, chkchykoly, chodchy, chykchy, chdchol, dchokchy, chokchy, chepchy, cheeetchol, choekcheey, chekchy, chekchy, chesokcho, chkorchy, cheekchey, chodchy, chchet, otchetchar, choetchaldy, chetalchs, chpchy, chchs, choepchey, chopchy, chepchy, chepchey, chetchody, chopchedy, chepcheesaly, chfaikchy, cheolchey, chpchedy, cheolchdy, chedchey, cheolchey, chchky, tchadchy, chchty, dorchedchey, cholchey, lchpchdy, checholtar, chopchedy, chedchy, chepchedy, chekchy, chofchdy, lkechch, chefchs, chedchs, chekchedy, chchdam, otodchch, chepchey, cheepchdy, pcheypchdar, chpchy, cholfchy, chefchy, chetchy, chepcheol, chorcholsal, chetchy, chchky, chtchy, cheokchey, ytchchy, schodchy, chorchor, chekchdy, chetchdy, chkchol, chokchor, chopcho, cheotchy, ykchochdy, chopcheey, chkchol, chchy, chokchey, cheoch, ytechypchy, pcholkchdy, chpchy, chechy, chepchy, cholcham, okchechy, okcheochy, polchechy, chotchs, ycheechy, cholkchy, otcheochy, chpcheor, chochs, chpchs, cheokcheo, chchy, tchedarchy, chefchedy, ycheochy, chtchy, chopchdy, chkchedaram, chpchy, cholchedy, chlchd, chopchy, cheokchy, chopchy, chokchedy, chotchedy, cheotchey, chdykchedy, cheolchear, okchdpchy, cheodchy, chokchy, cholcheey, kcholchdar, chepchedy, chctchy, chokchey, lchechdy, chepcham, chechey, cheolchal, pcheswchy, otchechy, cholchey, chokchy, chepchar, chopchey, chfchol, chtchol, chcheey, tcheolchy, tcheepchey, chpcheey, cheoltchedaiin, chokchedy, pchedchdy, chepched, chopcheod, chchdy, chotchdy, chokchaiin, chpchar, chotchdy, chkchol, chpcheod, pcholchdy, cholches, cheychear

ch, cth: 228
Words: chocthy, chcthy, chcthy, cholcthom, chcthy, chocthy, chocthy, chocthhy, ytchocthol, cheocthy, chcthod, pchocthy, chocthy, chcthy, chcthor, chcthy, qokchocthor, chcthody, pchocthy, chcthy, chcthy, chcthal, chcthy, chocthy, pchocthy, checthy, chcthy, chocthy, chcthy, chocthy, chocthy, chcth, chcth, chcthy, checthedy, cheecthy, chocthy, chocthy, cthorchy, chcthey, chctho, chocthey, checthy, checthy, dchcthy, chocthy, checthy, chcthedy, chcthdy, chcthy, chocthy, chocthy, chcthaiin, chcthy, chocthhy, cthochy, chcthy, checthhy, chocthy, chcthdy, chcthy, chcthy, chcthy, chcthy, chcthy, chcthy, chcthy, chcthy, chcthey, chctho, ychcthod, chocthody, ychocthy, cheocthy, schocthhy, chcthy, chcthy, chctho, chcthedy, choctheeey, odchecthy, chcthy, chcthhy, chcthy, chocthar, chcthhy, chcthy, chcthy, checthy, chcthedy, checthy, chcthy, chcthy, chcthedy, checthy, checthey, chcthy, chcthey, chcthy, chcthy, chcthy, chcthy, chcthy, chcthy, checthy, chcthhy, chcthy, schcthy, chcthy, chcthy, checthy, checthy, chcthy, checthy, checthy, checthy, checthy, chcthy, checthy, checthy, tchcthy, chcthy, chcthy, checthy, chcthy, chcthy, chcthey, chcthy, checthy, checthy, chcthdy, checthedy, chcthy, checthy, checthy, chcthy, olchcthy, chcthy, chcthy, checthy, cheocthey, chcthdy, chcthy, chcthy, chocthy, orchcthy, chcthdy, chocthy, chcthody, chcthy, chcthody, dcheecthey, chcthol, cheocthy, chocthey, chcthy, ychocthy, qochocthy, chcthy, chcthy, chctham, chcthykedy, chcthy, chcthory, chcthh, chocthey, chcthy, checthy, chcthy, chcthy, chcthy, chcthedy, chcthy, chocthy, chcthhy, chcthy, chcthy, chcthy, chcthy, chcthy, chectham, checthal, checthy, chcthy, checthey, checthdy, checthy, checthy, chcthedy, chcthydain, chcthar, chcthey, chctham, chcthy, chcthy, checthy, checthy, chcthy, chcthey, chcthy, chocthedy, chcthd, chcthy, chocthy, chcthal, fchoctheody, fchocthor, checthy, chcthy, chcthdy, chocthy, cheocthedy, chcthdy, chcthain, chcthey, chocthey, chcthed, cheocthy, cheocthy, checthy, chcthy, chcthy, checthey, chcthy, orchcthdy, chcthy, chcthy, chcthy

ckh, sh: 170
Words: shckhey, shockho, shckhy, shckhy, shckhol, shckh, shockhy, shckhy, shckhdy, shckhey, shockhhy, shckhy, qosheckhey, sheckhy, sheockhy, sheckhy, shckhefy, shckhy, orsheckhy, shckhy, ysheockhy, shockhy, shckhy, shckhdy, shckheody, shckhar, shckhed, shckhy, dsheckhy, ssheckhy, shckhy, shckhy, shckhy, sheckhy, sheckhy, sheckhy, sheckhy, lsheckhy, shckhy, shckhy, sheckhey, shckhy, shckhy, shckhy, shckhy, shckhy, shckhy, shckhey, sheckhy, sheckhy, shckhedy, sheckhedy, sheckhey, shckhedy, sheckhy, shckhy, shckhy, lshckhy, sheckhedy, shckhy, shckhy, lsheckhedy, sheckhdy, sheckhy, shckhey, sheckhy, sheckhy, sheckhy, sheckhy, sheckhy, shckhy, shckhy, sheckhy, lsheckhy, shckhy, sheckhedy, sheckhy, sheckhal, shckhy, shckhedy, shckhy, sheckhy, sheckhy, shckhey, sheckhy, sheckhy, sheckhy, sheckhdy, shckhey, shckhedy, shckhedy, shckhy, sheckhey, sheckhal, sheeckhy, shckhey, shckhy, shckhy, shckhy, shckhedy, shckhy, shckhy, shckhy, shckhy, shckhy, shckhdy, shckhor, shckhy, shckhy, roshckhy, shockhey, sholckhy, shckheo, sheockhy, shockhy, shockhey, sheockhey, ckhosh, shckheol, shockhey, sheofolckhhy, shockhy, sheockhey, shockhsy, shckhy, shckhy, shckhody, sheckhy, sheckhy, shckhy, shckhy, shckhy, shckhy, sheckhy, shckhy, shckhy, shckhy, sheckhy, sheockhey, shckhy, shckhy, shockhy, sheockhedy, sheckhy, shckhaiin, sheckhy, sheckhey, sheckhy, sheeckhy, shckhy, shckhey, shckhy, shckhey, sheckhy, shckhy, sheckhedy, shckhey, shckhey, shckhy, shockhol, sheckhy, shckhhy, shckhy, shckhy, shckhy, shckhy, sheckhy, shckhy, sheckhy, shckhy

ch, sh: 166
Words: shokcheey, chokoishe, shtchy, shytchy, shkchor, shokchy, shopchey, shotcho, shokchy, okcholksh, chshoty, wotchoshor, shkchy, shochy, chytshy, chokshor, shotchy, choldshy, okchodshy, fshodchy, chotshol, shachy, tcheoshy, shotchey, shoykcho, okchokshy, oschotshl, shepchy, chtshy, chokshy, shokchy, shytchy, chshy, shotchor, shorchey, otshcho, shepchdy, shefchdy, chdyshdy, shorchaiin, chshol, chshd, shekchy, shdchdy, chtshar, chekeesshy, shochol, cholsho, kechodshey, shtchy, shorpchor, choshy, chotshy, chotshol, shocho, shotchdy, tsholdchy, tchotshey, shopchy, sheetchy, shokchy, shekchy, shotchot, shofchy, sheeetchy, odshchol, shokchy, pchshy, shopalchedy, shchey, shekchy, shokchy, oshchey, chetshedy, qokchshy, chshek, shoekechy, shkechy, shkchodyal, shekchy, shopcho, otchosha, shchey, chotshy, shepchol, shopcho, chokshor, otchoshy, chewshees, choshydy, shepchy, chorshedy, shlches, chpsheedy, qolchshy, sharpchey, shepchedy, shepchy, sshkchdy, shchey, sheytchdy, shechy, sheekchy, pcholshedy, shefchdy, shetchy, sheopchey, lshechy, tshdch, sheekchey, sholfchor, okchshy, okshdchos, dshcheal, oepchksheey, ofchoshy, choshy, shdchy, cholshey, chosaroshol, cholsheody, sholkcheol, shekcheor, shchdy, cheolshy, shkchy, shorchdy, sheekchy, shdpchy, otechshy, lcheolshedy, cholkshedy, shokchey, shoefcheey, shedchy, sholfchdy, sheopchy, tcheorsho, shotchey, shotchy, shypchedy, shtchy, shchy, shechy, chetalshy, dchedshey, sheotchedy, shchy, chotshe, shchs, shokchy, cheshy, shkchedy, sholkeechy, sheolkchy, fcheshd, ykcheshd, choypshedy, chsho, kchedshedy, shkchedy, chkshy, shokchy, shtch, shekchy, shechol

cth, sh: 97
Words: shcthaiin, shocthol, shcthal, olshocthy, shcthhy, shcthy, shcthy, shocthy, shocthy, shcthy, shecthy, shocthy, shocthy, shcthy, shocthy, shocthy, shcthey, oshctho, shecthy, shcthy, shocthy, sheoctham, shcthy, sheocthy, shecthy, shcthy, shcthy, shcthey, shocthy, shocthhy, shecth, sheycthy, shocthy, shecthy, shecthy, shecthy, lshcthy, shcthy, shcthy, shecthy, shecthy, shcthedy, shcthey, shcthy, shecthy, shcthy, shecthy, shecthy, shcthy, shecthy, shecthy, shcthy, shcthy, shecthy, shcthhy, shecthy, shcthy, salshcthdy, sheecthey, shcthey, shcthy, shecthy, shcthy, shecthedy, shecthy, shcthy, shcthhy, shcthy, shcthy, shocthhy, shcthdy, shocthey, shctheo, cthoshol, shcthy, sheocthy, shcthhy, shcthy, shocthody, shocthy, shocthy, sheocthy, shcthy, shcthy, shcthy, shcthy, shcthy, shocthy, shecthey, shecthy, shecthy, shecthy, shcthy, shcthey, shcthy, shcthy, shcthy

ch, cph: 49
Words: chcphhy, chcphy, chcphy, chcphy, chcphy, chocphy, cphchar, chcphedy, pcheocphy, chocpheod, cphokchol, chcphol, chocphar, chocphy, ychocphy, chcpheor, chocphy, cheocphey, chcphhy, chcphy, chcphdy, chcphedy, chcphey, chcphey, checphedy, lchcphedy, checphey, chcphdy, chcphdar, sarchcphdy, chcphhy, chcphar, cheocphey, cheocphy, soefchocphy, chcphhdy, cheocphey, chcphy, chocphol, chcphy, checphey, chcphey, cphochy, chcphedy, chcphy, chcphy, chcphedy, chcpham, chcphy

sh, sh: 36
Words: shoshy, shoshy, shyshol, shotshy, shokshor, shtshy, shkshy, shosheky, shedshey, shekshey, shysho, sholkshy, sheshy, shodshey, ysholshy, shokeshy, shshy, otshshdy, sholshdy, sheshdy, shokshdy, sheepshey, shefshoro, shofshedy, shshdar, shetshdy, shekshol, psheodshy, shekshey, shotshey, psheessheeor, shctysh, sheshe, shealsho, sheekshy, lshalshy

cfh, ch: 24
Words: chcfhhy, chocfhey, checfhy, chcfhal, chcfhhy, cfhekchdy, chcfhhy, tdokchcfhy, olchocfhy, fchecfhy, chocfhy, chocfhy, chcfhdy, chcfhy, pchecfhey, fchcfhy, chocfhor, chcfhol, sochorcfhy, qopchcfhy, chcfhey, chcfhy, chcfhor, chocfhdy

cph, sh: 14
Words: shcphy, shocphhy, shecphy, shecphhedy, shecphy, shocphedy, shcphhy, shocphor, qoshocphy, sheocphy, shecphy, shcphoor, sheocphy, okeyshcphhy

cfh, sh: 6
Words: shcfho, shocfhey, shocfhy, shokocfhy, shcfhedy, shocfhy

ch, ch, sh: 4
Words: chokshchy, shetcheodchs, chlchpsheey, chpchshdy

ckh, cph: 3
Words: ckhhcphy, cpheckhy, qocpheeckhy

cph, cth: 3
Words: cphocthy, cphecthy, cpheocthy

ckh, ckh: 2
Words: ckhockhy, ckheckhy

ch, ch, cth: 2
Words: cheolchcthy, cholchecthody

ch, ckh, sh: 2
Words: shckhdchy, shckhchy

ch, ch, ckh: 2
Words: cheockhedchy, chckhyfchy

cfh, ckh: 1
Words: ckhcfhhy

ch, ch, ch: 1
Words: chopcheopchy

cth, sh, sh: 1
Words: ssholshecthy

ch, cth, sh: 1
Words: shecthedchy

cfh, ch, sh: 1
Words: psholpchcfhdy

cth, cth: 1
Words: cthoctho

ckh, cth: 1
Words: ckhocthy

Total sum of all substring combinations: 1522

I found a slight difference of 17 words between the increased "Total occurrence of the specified substrings" and "Total sum of all substring combinations:". I have no idea why.

[RobGea]

In this paper < Distinct word length frequencies: distributions and symbol entropies >  by Reginald Smith
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Thanks to Scarecrow, ninja xref: You are not allowed to view links. Register or Login to view. 

Smith says,

The literature on [..] Word length frequencies typically investigate the frequency of words of different lengths in syllables. 
These distributions are common amongst texts and are typically interpreted as a type of negative binomial distribution [..] or  [..] Hyper-Poisson { Displaced Poisson distribution }.

Further in  "Section 2. Distinct word length distributions."
Figure 1.  is a  "Graph of the frequency of distinct words by word length"  for various languages,

where English shows a distribution similar to that of English in Stolfis' Figure 2. [1]
and curiously the language graphs for ES, PT, IT, FR, DE, RU. ( to my eye at least ) show a visual similarity to a binomial distribution.
     .
[1]  Web archive link to Stolfi page "On the VMS Word Length Distribution"    because unicamp link is currently not accessible.
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[lelle]

Can anyone please explain what might be a reasonable underlying mechanism for the observed word length distribution in the VMs? The author flipping 9 (or 10) coins simultaneously and counting numbers of heads. Numbers of heads will then dictate the length of the current word?

[bi3mw]

Can anyone please explain what might be a reasonable underlying mechanism for the observed word length distribution in the VMs? The author flipping 9 (or 10) coins simultaneously and counting numbers of heads. Numbers of heads will then dictate the length of the current word?

Coin toss as a model for the letters of a word ( as I understood it ):

The binomial distribution only knows two "states": success or failure.
With n = 9, a coin is therefore tossed 9 times in succession, with heads standing for success and numbers for failure.

Each coin toss is seen as a decision as to whether another letter is added to the word.
On each of the "n" (here 9 or 10) tosses, the coin (or the probability "p", here symmetrical 0.5) influences how long the word becomes.
"Heads" could mean, for example, that a letter is added.
"Number" could mean that no further letter is added.
The binomial distribution then models the probability of how many letters are contained in a word at the end of such a process.

Stolfi`s "Shifted by 1" means that the actual word length does not range from 0 to 9, but from 1 to 10. Each value from the binomial distribution is therefore increased by 1. A word with a length of 0 should not occur.

In my code, as soon as the target word length has been determined, the words are shortened or lengthened accordingly.

[Koen G]

Wouldn't the most likely explanation be that whatever system was used resulted in the same distribution as if coins had been tossed, rather than actually tossing 9 coins (or a coin 9 times) before writing a word?

[bi3mw]

Wouldn't the most likely explanation be that whatever system was used resulted in the same distribution as if coins had been tossed, rather than actually tossing 9 coins (or a coin 9 times) before writing a word?

Yes, you are right. After all, we don't know how this distribution came about in the VMS. We can only speculate about that. I was thinking more about the creation of binomially distributed text from normal text.

[ReneZ]

Wouldn't the most likely explanation be that whatever system was used resulted in the same distribution as if coins had been tossed, rather than actually tossing 9 coins (or a coin 9 times) before writing a word?

In modern times, yes, but in the middle ages?

There is some thought about the binomial distribution in this paper:
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