28
vs. (3) ʕProbability ͱԿΛҙຯΔͷʕ Տ ܟ38 3 ෦ଔ ۀ40 मऴ લճͷͷ ߘ([17])ʢҎԼɼͷҎલͷߘΛ୯ʹʮલճʯɼʮલʑճʯͱͱʹ ΔʣΛʹΓͱΖɼΖΖ༗ӹͳίϝϯτΛɽ·ɼͷෆ᪾ ͳʹஸೡʹରԠෳͷ༑ਓݾͷʑʹ·ޚਃɽճ ͷߘͰఠίϝϯτΛʹߟɽػΒஶ ݖ࡞ޢͷظݹݙΤϒʹ։ΕɼશจΛμϯϩʔυͰΔ߹ ΔɽΕΒΛরΔͱʹΑલճΒͷཧղਐΜ෦Δɽճલճ ߟෆͱΖΛɼॳʹҙɼຊʹͱͷ probability ೦༁ޠΊఆணܦҢͷੳɼհΛऴͱɼ ޚঝͷΑͳࠓࡢͷ৽ܕίϩφɾΠϧεಈͰେਤॻͷར༻େ෯ʹݶΕΑʹจݙΛௐΔͱग़དྷͳɽͷΊɼճߟͰͳͱΖ ɼग़དྷΒҰճνϟϯεΛճʹͱୈͰΔɽ 1 લճͷ 1.1) લճɼ 67 ทͷ٭Ͱɼʮ౻ᖒརتଠͷʰੜอၤʱ (1889,M22,[9]) ɼυɾϞϧΨ ϯ 1 (De Morgan) ͷɼ An Essay on Probabilities, and Their Application to Life Contingencies and Insurance Offices. (1838, ఱอ 9,[7]) ͰͳͱߟΒΕΔɽผදΔ༧ ఆͰΔɼ༰ճʹհΔΓͰΔɽʯͱॻͷɼ RIMS( ཧղੳڀݚ) ͷ։ڀݚܕձʮͷڀݚʯͰදΔػձΛಘɽɼੜอ ݥʹશͷυૉਓͳͷͰɼલ·Ͱڭͷ٬һڭतΛΕΞΫνϡ ΞϦʔͷதૉੜ2 ͷશ໘తڠΛಘɼಡͷڞஶจɼʮ౻ᖒརتଠஶʰੜ อᯃʱʹΈΔͷձݙߩͷΓʯɽRIMSKˆokyˆ uroku Bessats, (2020,B81,33-52) ͱදͰɽʹطΤϒͰ։ΕΔͱͷͰɼΒর ɽͳɼલճͷ༰ͷओཁ෦क़େʹ։Εୈ 30 ճγϯ ϙδϜʹɼʮ܉Կ ނprobability Λͱ༁ͷʕͻͱͷԾઆʕʯͱ ͰදΒɽจষԽͷΕक़େͷ HP ʹ։ΕΔͷͰ ΒʹߟΕͰΔɽ 1.2) ʮʯͷಡΈʹɽͷςϨϏχϡʔεͰɼΔॻΛҾ ʮΜʯͱग़ΔΒɼԿͷʹʮk¯osanʯͱԻΔͷͱ৴ ͷɼΔਤॻͰۮ. ʮ ӳӳɼฌޠ,ʯ (1944,S19,[45]) ͱখͳ Λݟ probability ͷ༁Λݟڼఱɽʹ k¯ozan ɼͱॻΕͷͰ 1 υɾϞϧΨϯͷͱߍߴͷڭՊॻʹग़Δ൴ͰΔɽ 2 ࡏݱಉ૭ձͷΛۈΊΒΕΔɽ 寄稿 68

Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

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Page 1: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

vs. (3)

Probability

38 3 40

([17])

probability

1

1.1) 67 (1889,M22,[9])1(De Morgan) An Essay on Probabilities, and Their Application to Life Contingencies

and Insurance Offices. (1838, 9,[7])

RIMS(

)

2

RIMS Kokyuroku Bessats, (2020,B81,33-52)

30

probability

HP

1.2)

kosan

. , (1944,S19,[45])

probability kozan

1

2

1

寄稿

―68―

Page 2: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

Kozan probability, chance dahi

( ) 3

gosa a probable error

4

1.3)

(2009,H21,[1]) (87 -88 )

-

1.4)

10(1877)

(2016,H28,[44])

3 [3],1474

2

―69―

Page 3: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

a) :

(1850( 3)–1935( 10)) 1886( 19)5

probability

b) : 1873( 6)

1884( 17)

1877( 10)

6

2,30

1991(

3)

(466 –467 ) 1900( 33) 5 1)

2)

([17],65 ) 1887( 20)7

5

6 ([18])

7

3

―70―

Page 4: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

8

1.5) probability

2 5

9 probability Wahrscheinlichkeit

Wahrscheinlichkeitsfunction Wahrscheinlichkeitsrehnung

probability (85 )

1866( 2) 1913( 2) 4

Wahrscheinlichkeit

45

6

probability

probability

3

10

8 A. 63

9

10 18([21])

4

―71―

Page 5: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

probability

1 15

4 3 21

10 20

2 probability

2.1

probability 10

1 15(1882) 4 ([38])

probability

76 )

(probable

error)

2 16(1883) ([30])

probability

probability ([2],187 ) 1893(

26)

3 21(1888) ([39]) probability

probability

1

4 1

5

―72―

Page 6: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

([3],137 –140 )

11

12

3 4 1/3

4 3

3

4.2

Liagre ([27])

4 24(1891) ([15])

3

( [3],145 )

([3],145

–150 )

5 27(1894) : 5([6])

25(1892)

6

([37], 172 –177 ) ([42]) ([3],158

–166 ) ([3],158 )

11

([42],16 )

12

6

―73―

Page 7: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

26 6

([42] 350 )

6 28(1895) 13 Calculus of Probability and Method of

Least Squares.

probability

3

3 3

20 5

7 ([2],184 –188 [3],143 –154 ) 34(1901)

(4 )

39 8

36 8

11 21

8 34(1901)

14 4([15]) 12

4

24 4

9 35(1902) ([11])

13

14 pdf

7

―74―

Page 8: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

(25 )

G. Hagen: (1837, 8,[10]). Grundzge der Wahrscheinlichkeitsrechnung. Berlin:Dmmler.

2 1867, 3 3 1882, 5

10 41(1908) ([12])

3

21

([2],187 )

(185 ) 36

42 1

La Science et L’hypothese probability

probability

5 5.1) ([3], 180 –181 ) J. Bertrand Calcul des

Probabilites 1889( 22)

Bertrand

(52 –56 ) Bertrand (p.72–p.76)

Bertrand Probabilites Totales

8

―75―

Page 9: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

3

Bertrand

11 15 16([31]).

2

6

9

1. :(1939, 14,[28]). 17.

2. :(1955, 30,[29]). .

15 (1890(M33)–1961(S36)) (1940,S15,[42])

16

( 1 6 ) ([35])15 6

([42]) 20 3

probability

17

9

―76―

Page 10: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

([3],135 )

1.

S.F. Lacroix:(1864, ,[25]).Traite elementaire du calcul des probabilites.

2.

H. Laurent:(1873, 6,[26]).Traite du calcul des probabilites.

3.

J.B.J. Liagre:(1879, 12,[27]).Calcul des probabilites et theorie des erreurs avec des

applications aux sciences d’observation en general et la geodesie en particulier.

2.2

([14])

probability

18

probability

probability

3

18

(2013,H25,[40])(2018,H30,[41])

10

―77―

Page 11: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

(2000,H12,[2])

(2012,H24,[3]) (2000,H12,[19]—

2010,H22,[24]) ([3])

3 ([3],136 -137 )

1) 3 Liagre ([27])

2)

(1856, 3–1893, 26)

3) (1847, 4–1931, 6)

3

3

3( ) 6

6( )

3.1

19

1

19

11

―78―

Page 12: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

3( )20 4

([3],145 )

3( )

7 34 ([2],184 )

4

([2],185 ) 23 12 1 34 11 2

4

21

4 722

25,6

2 11 (1893(

26) ) 23 26

7 34

26

3( 21 )

4( 24 )

([2], 185 ) 36 39

3( ) 1

20 ([39])

21[42] (349 )

22 ([3],179 )23 C10060306200

12

―79―

Page 13: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

8 4 4

7 ([2], 184 –187 )

a-1)

3( ) 4

45 ([36])

a-2)

a-3) 36 4 8

3

(185 )

8

8 10

9

9

1,3,4

Hagen ([10])

HP

1903( 36 )(C07060281000)

13

―80―

Page 14: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

([2], 185 )

36 4

8 9 (1901( 34) )(C09122689800)

(1912,M45,[36])

24

4

([3])

22

45

1

24 Summary Reports on the Teaching of Mathematics inJapan

14

―81―

Page 15: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

(22 )

(12 ) (15 )

([2])

probability

3( )

21

3( ) 15 1( 4)

21

3( )

3.2

3.2-1) 1( 15 ) 1

20

tgY =x

ytg tangent x =

ytgβ

sinα

3

15

―82―

Page 16: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

2

ε ( )25

µ

r ( )1

2

r = 0, 6745.ε, r = 0, 8453.µ

2

ε µ

3 ) 1/2

1/2 1

1

. ,26

3 )

( [39], (2),60 )

3( )

3.2-2) ( ) m! |m 1 · 2 · 3 · · ·m3

mCn (p+ q)m

Cnn

27 3 29 mCn =m(m− 1)(m− 1) . . . (m− n+ 1)

1.2.3. . . . n

25

26 K27 3 mCn 59 ,60 Cn

m

Cnm

16

―83―

Page 17: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

2 6( ) Cnm

4 6( ) 9(30 )

|m 95 m!

3-2-3) ( ) ℓ.([39],((1),57 ,(2),62 ) L.([39],(3),

144 ,145 ) 4(54 ) L.

30 log.

log

5 5.1)

J.Bertrand (1889, M22,[5]) ℓ2 = 0, 6931(p.13)

6( ) 1895( 28) log

( 16 ) Bertrand ℓ.

4 3( )

2.2 ([3],136 -137 ) 3( )

3 Liagre ([27],1879,M12)

Calcul des probabilites et theorie des erreurs avec des applications aux sciences d’observation

en general et la geodesie en particulier.

Liagre

( 2 ) pdf

Liagre ( [3],136 )

1

600

3( ) 40

4.1 3( )

3( ) Liagre

3( ) 28 ( )29

28 ([3],137 –138 )29 ([3],138 )

17

―84―

Page 18: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

(1 ) (1 ) (1 )

(1 )

30(1.5 ) (p+ q)m (4 )

(3.5 ) p = q =1

2(2 )

(3 ) (2 )

(1 ) (0.5 )

x (3 )

(2 )

(2 )

(1.5 )

(1 ) t ±αh (0.5 )

(2 )

( ) x′, x′′, x′′′, · · · , xp

(0.5 )

(1.5 ) (2 )

(1.5 )

(1.5 )

(0.5 )

(1 )

(1 )

43

4.2 Liagre([27]) 3( )([39])

3( )( ) Liagre 9 §116.Erreurmoyenne de l’unit de poids (p.298)

Liagre

30 ([3],137 )

18

―85―

Page 19: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

40 1

4.2.1)

4 Liagre I II 62

1/6

2

4 6

A1, . . . , An

P (∪nAn))

A1, . . . , AnP (A1)∑n P (An)

A1

A,B P (A∩B)

A A la probabilite

absolue, relative, composee, simple la probabilite absolue

Liagre La probabilite d’un evenement qui

correspond l’arrivee de plusieurs eventualites, est la somme des probabilites relatives a

chacune de ces eventualites. la probabilite relative la

probabilite absolue 6(

) The principle of Total Probability

2 Laurent 47 Principe de la probabilite totale

4.2,8) 5

5.1)

Liagre

probabilite a priori(Liagre ) 4

probabilite a posteriori

4

4.2.2) 2 7 8

Liagre p.44

4.2.3) 10 m

n m′ n′

Liagre p.45–p.46

19

―86―

Page 20: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

4.2.4) 4.2.3) ,

Liagre p.48–p.49

Liagre

6( )

4.2.5) 11 Liagre

III (p.63–p.110)§23 §39§

Liagre la formule

de Stirling31

32

Liagre

§36. Loi des grands nombres

10

4.2.6) Liagre

p.91 p.92

Liagre

Liagre

4.2.7) Liagre

9 §116.p.298Liagre

Liagre

31

32Central limit theorem 1920 G.Polya. ([8])

20

―87―

Page 21: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

3( ) Liagre ([27])

4.2.8) 1 2 ([3],136 ) 1 Lacroix

2 2 7 8

Liagre Lacroix

Lacroix

D.

([3],136 ) 2 Laurent

IV Methode des moindres

carres(method of least squares)

3( )

I

:

nn

1 · 2 · 3 · · ·n=

1

∫ +π

−πen(cos t+

√−1 sin t)−nt

√−1dt.

{cosnt, sinnt;n = 0, 1, 2, · · ·}

4.2.1) Liagre

6( ) The principle of Total

Probability Principe de la probabilite total Laurent

47

Cnm 3( )

5 3( )

3( )

6( ) 10

4,9,11

21

―88―

Page 22: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

16(1883) 19(1886) ( 4

) 2033

10

34 1881( 14)

([4], 96 ) 1886( 19) 10

6

3( ) 2.2

2) 3) 3)

5.1) 2)

1 1878( 11)

1886( 19)

Liagre

3( )

Liagre

Liagre

Liagre

4.2.1) 4.2.8)

6 the principle of Total Probability

the principle of compound probability principle

(1855, 2–1923, 12)

1 1878( 11)

1883( 16)

[23],240 )

33 1 23 ([43])341856( 3)–1933 8). 3 ( )

22

―89―

Page 23: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

Bertrand

3( )

Bertrand Calcul des probabilites(1889,M22)([5])

([34],85 ) II

PROBABILITES TOTALES ET PROBABILITES COMPOSEES

22 3( ) Bertrand35 3( )

Liagre

probability

5.2) 3)

2)

5 ([6])

([42],352 )

( )

D. 36

37

1889( 22) 38

3( )

35 ([33], 95 ) ([3], 3836David Murray(1830–1903).

1873( 6) 637 7 0148810038 : C09060110800

23

―90―

Page 24: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

(1991,H3,[16],274 )

326 )

([23],233 )

([16],327 )

OS

8

([42],343 ) 39

( )

1990( 23) 12

3( 21 )

([3],140 ) 3( )

21

39 2 ( ) 1880( 13 )

24

―91―

Page 25: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

Liagre

1 15 4

probability

1 4

([3],81 – 84 )

3

10

probability

6

3

100 ([32],126

)

probability

1 4( 15 )

4

3

1

3

100

3

2,6,10

1 3 1

3( )

3

25

―92―

Page 26: Probability q x ? ¯ b w T - math.kyoto-u.ac.jp · and Insurance Offices. (1838, 1 -9,[7]) p x s M T q ß Q } Ä I x M C ¯ b ' p K U | º 0 x Í s t º p b m p K } ¯ q { M h w i

3(1888,M21) 2(1955,S30)

14 1

3 2140

6

5 5.1)

3( )

2.2

3( )

[1] : 2009(H21). .

[2] : 2000(H12).1130 , 174–188.

[3] ————-: 2012(H24). .

[4] : 1937(S12). .

[5] J. Bertrand: 1889(M22). Calcul des Probabilites. Gauthier-Villars.

[6] : 1894(M27). 5 .

https://dl.ndl.go.jp/search/searchResult?searchWord=%E7%A0%B2%E5%A4%96%E5%BC%

BE%E9%81%93%E5%AD%A6&featureCode=all&viewRestrictedList=0&tocItemId=info%

3Andljp%2Fpid%2F844791

[7] A. De Morgan: 1838( 9). An Essay on Probabilities and on Their Application to LifeContingencies and Insurance Offices. London. Orme, Brown, Green & Longmans.

https://archive.org/details/anessayonprobab00morggoog/page/n8

[8] H. Fischer: 2010(H22). A History of the Central Limit Theorem: From Classical to ModernProbability Theory. Springer Science & Business Media.

https://books.google.co.jp/books?id=v7kTwafIiPsC&printsec=frontcover&hl=

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[34] ———-: 1947(S22). 3 .

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