Law of large numbers

The law of large numbers, or LLN for short,^[1] is a theorem from statistics. It states that if a random process is repeatedly observed, then the average of the observed values will be stable in the long run. This means that as the number of observations increases, the average of the observed values will get closer and closer to the expected value.

For example, when rolling dice, the numbers 1, 2, 3, 4, 5 and 6 are possible outcomes. They are all equally likely. The population mean (or "expected value") of the outcomes is:

(1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5.

The following graph shows the results of an experiment of rolls of a die. In this experiment, it can be seen that the average of die rolls varies wildly at first, but as predicted by the LLN, the average stabilizes around the expected value of 3.5 as the number of observations become large.

History

Jacob Bernoulli first described the LLN. He said the concept was so simple that even the stupidest man instinctively knows it is true.^[2] Despite this, it took him over 20 years to develop a good mathematical proof. Once he had found it, he published the proof in Ars Conjectandi (The Art of Conjecturing) in 1713. He named this his "golden theorem". It became generally known as "Bernoulli's theorem" (not to be confused with the law in physics with the same name). In 1835, S.D. Poisson further described it under the name "la loi des grands nombres" (the law of large numbers).^[3] Thereafter, it was known under both names, but the "law of large numbers" is most frequently used.

Law of large numbers of Googol to a Maximusmultillion (limit)
Googol: 10^100
Googolchime: 10^1,000
Googoltoll: 10^10,000
Googolgong: 10^100,000
Maximusmillion: 10^1,000,000
Goospolplex: 10^10,000,000
Googolbong: 10^100,000,000
Maximusbillion: 10^1,000,000,000
Trialogue: 10^10,000,000,000
Googolthrong: 10^100,000,000,000
Maximustrillion: 10^1,000,000,000,000
Trdoocolplex: 10^10,000,000,000,000
Googolgandingan: 10^100,000,000,000,000
Maximusquadrillion: 10^1,000,000,000,000,000
Maximusbyllion: 10^10,000,000,000,000,000
Googolquintigong: 10^100,000,000,000,000,000
Maximusquintillion: 10^1,000,000,000,000,000,000
Guppyplexichunk: 10^10,000,000,000,000,000,000
Guppyplex: 10^100,000,000,000,000,000,000
Maximussextillion: 10^1,000,000,000,000,000,000,000
Maximusseptillion: 10^10^24
Minnowplex: 10^10^25
Maximusoctillion: 10^10^27
Maximusnonillion: 10^10^30
Maximustryllion: 10^10^32
Maximusdecillion: 10^10^33
Gobyplex: 10^10^35
Maximusundecillion: 10^10^36
Maximusduodecillion: 10^10^39
Maximustredecillion: 10^10^42
Maximusquattuordecillion: 10^10^45
Maximusquindecillion: 10^10^48
Gogolplex: 10^10^50
Maximussexdecillion: 10^10^51
Maximusseptendecillion: 10^10^54
Maximusoctodecillion: 10^10^57
Maximusnovemdecillion: 10^10^60
Maximusvigintillion: 10^10^63
Prawn-plex: 10^10^65
Maximusunvigintillion: 10^10^66
Maximusduovigintillion: 10^10^69
Maximustrevigintillion: 10^10^72
Lightweight-plex / Maximusquattuorvigintillion: 10^10^75
Maximusquinvigintillion: 10^10^78
Ogolplex: 10^10^80
Maximussexvigintillion: 10^10^81
Maximusseptenvigintillion: 10^10^84
Twerpuloid-plex: 10^10^85
Maximusoctovigintillion: 10^10^87
Maximusnovemvigintillion: 10^10^90
Maximustrigintillion: 10^10^93
Googolplex: 10^10^100
Googoltyplex: 10^10^101
Eleventyduplex: 10^10^110
Maximusquadragintillion: 10^10^123
Maximusquinquagintillion: 10^10^153
Maximussexagintillion: 10^10^183
Gargoogol-plexed: 10^10^200
Maximusseptuagintillion: 10^10^213
Maximusoctogintillion: 10^10^243
Maximusnonagintillion: 10^10^273
Ecetonplex / Maximuscentillion: 10^10^303
Googolplexiding: 10^10^500
Maximusducentillion: 10^10^603
Maximustrecentillion: 10^10^903
Googolplexichime: 10^10^1,000
Maximusquadringentillion: 10^10^1,203
Maximusquingentillion: 10^10^1,503
Maximussescentillion: 10^10^1,803
Maximusseptingentillion: 10^10^2,103
Maximusoctingentillion: 10^10^2,403
Maximusnongentillion: 10^10^2,703
Maximusmillillion: 10^10^3,003
Googolplexibell: 10^10^5,000
Maximusdumillillion: 10^10^6,003
Maximustrimillillion: 10^10^9,003
Googolplexitoll: 10^10^10,000
Maximusquadrimillillion: 10^10^12,003
Maximusquinmillillion: 10^10^15,003
Maximussexmillillion: 10^10^18,003
Maximusseptimillillion: 10^10^21,003
Maximusoctimillillion: 10^10^24,003
Maximusnonimillillion: 10^10^27,003
Maximusmyrillion / Maximusdecimillillion: 10^10^30,003
Googolplexigong: 10^10^100,000
Maximuscentimillillion: 10^10^300,003
Milliduplexion: 10^10^1,000,000
Maximusmicrillion: 10^10^3,000,003
Googolplexibong: 10^10^100,000,000
Maximusnanillion: 10^10^3,000,000,003
Googolplexithrong: 10^10^100,000,000,000
Maximuspicillion: 10^10^3,000,000,000,003
Maximusfemtillion: 10^10^(3*10^15+3)
Maximusattillion: 10^10^(3*10^18+3)
Maximuszeptillion: 10^10^(3*10^21+3)
Maximusyoctillion: 10^10^(3*10^24+3)
Maximusxonillion: 10^10^(3*10^27+3)
Maximusvecillion: 10^10^(3*10^30+3)
Maximusmecillion: 10^10^(3*10^33+3)
Maximusduecillion: 10^10^(3*10^36+3)
Maximustrecillion: 10^10^(3*10^39+3)
Maximustetrecillion: 10^10^(3*10^42+3)
Maximuspentecillion: 10^10^(3*10^45+3)
Maximushexecillion: 10^10^(3*10^48+3)
Maximusheptecillion: 10^10^(3*10^51+3)
Maximusoctecillion: 10^10^(3*10^54+3)
Maximusennecillion: 10^10^(3*10^57+3)
Maximusicosillion: 10^10^(3*10^60+3)
Maximusmeicosillion: 10^10^(3*10^63+3)
Maximusdueicosillion: 10^10^(3*10^66+3)
Maximustrioicosillion: 10^10^(3*10^69+3)
Maximustetreicosillion: 10^10^(3*10^72+3)
Maximuspenteicosillion: 10^10^(3*10^75+3)
Maximushexeicosillion: 10^10^(3*10^78+3)
Maximushepteicosillion: 10^10^(3*10^81+3)
Maximusocteicosillion: 10^10^(3*10^84+3)
Maximusenneicosillion: 10^10^(3*10^87+3)
Maximustriacontillion: 10^10^(3*10^90+3)
Googolplexian: 10^10^10^100
Maximusgoogolillion: 10^10^(3*10^100+3)
Googoltyduplex: 10^10^10^101
Maximustetracontillion: 10^10^(3*10^120+3)
Maximuspentacontillion: 10^10^(3*10^150+3)
Maximushexacontillion: 10^10^(3*10^180+3)
Maximusheptacontillion: 10^10^(3*10^210+3)
Maximusoctacontillion: 10^10^(3*10^240+3)
Maximusennacontillion: 10^10^(3*10^270+3)
Maximushectillion: 10^10^(3*10^300+3)
Maximusdohectillion: 10^10^(3*10^600+3)
Maximustriahectillion: 10^10^(3*10^900+3)
Maximustetrahectillion: 10^10^(3*10^1,200+3)
Maximuspentahectillion: 10^10^(3*10^1,500+3)
Maximushexahectillion: 10^10^(3*10^1,800+3)
Maximusheptahectillion: 10^10^(3*10^2,100+3)
Maximusoctahectillion: 10^10^(3*10^2,400+3)
Maximusennahectillion: 10^10^(3*10^2,700+3)
Maximuskillillion: 10^10^(3*10^3,000+3)
Maximusvecekillillion: 10^10^(3*10^30,000+3)
Maximushectekillillion: 10^10^(3*10^300,000+3)
Maximusmegillion: 10^10^(3*10^3,000,000+3)
Maximusvecemegillion: 10^10^(3*10^30,000,000+3)
Maximushectemegillion: 10^10^(3*10^300,000,000+3)
Maximusgigillion: 10^10^(3*10^3,000,000,000+3)
Maximusterillion: 10^10^(3*10^3,000,000,000,000+3)
Maximuspetillion: 10^10^(3*10^(3*10^15)+3)
Maximusexillion: 10^10^(3*10^(3*10^18)+3)
Maximuszettillion: 10^10^(3*10^(3*10^21)+3)
Maximusyottillion: 10^10^(3*10^(3*10^24)+3)
Maximusxennillion: 10^10^(3*10^(3*10^27)+3)
Maximusdakillion: 10^10^(3*10^(3*10^30)+3)
Maximushendillion: 10^10^(3*10^(3*10^33)+3)
Maximusdokillion: 10^10^(3*10^(3*10^36)+3)
Maximustradakillion: 10^10^(3*10^(3*10^39)+3)
Maximustedakillion: 10^10^(3*10^(3*10^42)+3)
Maximuspedakillion: 10^10^(3*10^(3*10^45)+3)
Maximusexdakillion: 10^10^(3*10^(3*10^48)+3)
Maximuszedakillion: 10^10^(3*10^(3*10^51)+3)
Maximusyodakillion: 10^10^(3*10^(3*10^54)+3)
Maximusnedakillion: 10^10^(3*10^(3*10^57)+3)
Maximusikillion: 10^10^(3*10^(3*10^60)+3)
Maximusikenillion: 10^10^(3*10^(3*10^63)+3)
Maximusicodillion: 10^10^(3*10^(3*10^66)+3)
Maximusictrillion: 10^10^(3*10^(3*10^69)+3)
Maximusicterillion: 10^10^(3*10^(3*10^72)+3)
Maximusicpetillion: 10^10^(3*10^(3*10^75)+3)
Maximusikectillion: 10^10^(3*10^(3*10^78)+3)
Maximusiczetillion: 10^10^(3*10^(3*10^81)+3)
Maximusikyotillion: 10^10^(3*10^(3*10^84)+3)
Maximusicxenillion: 10^10^(3*10^(3*10^87)+3)
Maximustrakillion: 10^10^(3*10^(3*10^90)+3)
Googolplexianth: 10^10^10^10^100
Maximustekillion: 10^10^(3*10^(3*10^120)+3)
Maximuspekillion: 10^10^(3*10^(3*10^150)+3)
Maximusexakillion: 10^10^(3*10^(3*10^180)+3)
Maximuszakillion: 10^10^(3*10^(3*10^210)+3)
Maximusyokillion: 10^10^(3*10^(3*10^240)+3)
Maximusnekillion: 10^10^(3*10^(3*10^270)+3)
Maximushotillion: 10^10^(3*10^(3*10^300)+3)
Maximusbotillion: 10^10^(3*10^(3*10^600)+3)
Maximustrotillion: 10^10^(3*10^(3*10^900)+3)
Maximustotillion: 10^10^(3*10^(3*10^1,200)+3)
Maximuspotillion: 10^10^(3*10^(3*10^1,500)+3)
Maximusexotillion: 10^10^(3*10^(3*10^1,800)+3)
Maximuszotillion: 10^10^(3*10^(3*10^2,100)+3)
Maximusyootillion: 10^10^(3*10^(3*10^2,400)+3)
Maximusnotillion: 10^10^(3*10^(3*10^2,700)+3)
Maximuskalillion: 10^10^(3*10^(3*10^3,000)+3)
Maximusdakalillion: 10^10^(3*10^(3*10^30,000)+3)
Maximushotalillion: 10^10^(3*10^(3*10^300,000)+3)
Maximusmejillion: 10^10^(3*10^(3*10^3,000,000)+3)
Maximusgijillion: 10^10^(3*10^(3*10^3,000,000,000)+3)
Maximusastillion: 10^10^(3*10^(3*10^(3*10^12))+3)
Maximuslunillion: 10^10^(3*10^(3*10^(3*10^15))+3)
Maximusfermillion: 10^10^(3*10^(3*10^(3*10^18))+3)
Maximusjovillion: 10^10^(3*10^(3*10^(3*10^21))+3)
Maximussolillion: 10^10^(3*10^(3*10^(3*10^24))+3)
Maximusbetillion: 10^10^(3*10^(3*10^(3*10^27))+3)
Maximusglocillion: 10^10^(3*10^(3*10^(3*10^30))+3)
Maximusgaxillion: 10^10^(3*10^(3*10^(3*10^33))+3)
Maximussupillion: 10^10^(3*10^(3*10^(3*10^36))+3)
Maximusversillion: 10^10^(3*10^(3*10^(3*10^39))+3)
Maximusmultillion: 10^10^(3*10^(3*10^(3*10^42))+3) (limit)
etc.

Other mathematicians also contributed to make the law better. Some of them were Chebyshev (who proved a more general version of the law for averages^[4]), Markov, Borel, Cantelli and Kolmogorov. After these studies, there are now two different forms of the law: One is called the "weak" law, and the other the "strong" law.^[5] These forms do not describe different laws. They have different ways to describe the convergence of the observed or measured probability to the actual probability. The strong form of the law implies the weak one.

Law Of Large Numbers Media

Illustration of w:Law of large numbers.

Related pages

Central limit theorem

References

↑ List of Probability and Statistics Symbols (in en-US). Math Vault (2020-04-26). Retrieved 2020-10-14.
↑ Jakob Bernoulli, Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis, 1713, Chapter 4 (Translated into English by Oscar Sheynin)
↑ Hacking, Ian. (1983) "19th-century Cracks in the Concept of Determinism"
↑ Law of large numbers | statistics (in en). Encyclopedia Britannica. Retrieved 2020-10-14.
↑ Law of Large Numbers. www.probabilitycourse.com. Retrieved 2020-10-14.

[1] List of Probability and Statistics Symbols (in en-US). Math Vault (2020-04-26). Retrieved 2020-10-14.

[2] Jakob Bernoulli, Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis, 1713, Chapter 4 (Translated into English by Oscar Sheynin)

[3] Hacking, Ian. (1983) "19th-century Cracks in the Concept of Determinism"

[4] Law of large numbers | statistics (in en). Encyclopedia Britannica. Retrieved 2020-10-14.

[5] Law of Large Numbers. www.probabilitycourse.com. Retrieved 2020-10-14.

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