x > 0, log 10 x = ln x

(1)

Épreuve de setion européenne

Benford'slaw

Thelesontheharddiskofanyomputerhavevarioussizes(inKb). Ifwetakealookatthem,

weanounthowmanynumbersbeginwith1,howmanybeginwith2,howmanybeginwith3,

andsoon. Youmightexpetthattherewouldbethesameproportionofnumbersbeginningwith

eahdierentdigit,roughly1/9,butitisverylikelythatyouwillbewrong!

Forexample,thetablebelowistheresultofanexperimenton150,000lesofMyles folder

onChristmasEve,2010:

Firstdigit 1 2 3 4 5 6 7 8 9

Numberof les 48552 23923 16407 12989 14364 10073 8884 7749 7059

Relativefrequeny .324 .16 ? .087 .096 .067 .059 .052 .047

Surprisingly, asfor many kindsof data, the distribution of rstdigits is highly asymmetri,

the most ommon digit being 1and the least ommon 9. This fat was disovered in 1881 by

the Amerian astronomerSimon Newomb, by notiing that in logarithm books (used at that

timeto performalulations)theearlierpages weremuhmorewornout 1

thantheother pages.

Thephenomenonwasredisoveredin1938byFrankBenford,aphysiistattheGeneralEletri

Company,wholaimedthattherelativefrequenyofnumbersthatstartwiththedigit

D

^should

be:

^where, ^for^all

^is^the^deimal^logarithm^of

Benford'slawisusedtotrakdownfraudin variousdomains;ithasbeeninvokedasevidene

of fraud in the 2009 Iranian eletions. In June 2010, onsultants working for politial website

Daily Kos used Benford's law, among other tools, to ndserious awsin thedata olleted by

pollingompanyResearh2000(R2K).Thisledto thetermination ofR2K'sontratwithDaily

Kos...

Adaptedfromvarioussoures,(TedHill'swebsite,Wikipedia,Plus magazine).

Questions

1. Whydidn't the2010Christmastabletakeinto aountthedigit0?

2. Explain howNewomb disoveredtheFirstdigitphenomenon.

3. Computetherelativefrequenymissinginthetable.

4. Using Benford's formula, omputethe theoretial frequenyfor eah digit. Doesthe 2010

Christmasomputerexperimentmaththeriteria?

5. Should you useBenford's lawto hooseyourlotterynumbers? Would youuseit with the

ageofpolitiiansintheFrenhAssembléeNationale? Explainyouranswers.

6. Desribeasimpleargumentin favorofusingBenford'slawtotraedownfraud.

2011-08Benford'slaw

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