De Moivre and Normal Distribution

The First Central Limit Theorem

Central Limit Theorem

• CLT states that, given certain conditions, the mean

of a sufficiently large number of independent rando

m variables, each with finite mean and variance, will

be approximately normally distributed.

• De Moirve-Laplace theorem is a special case of the

CLT, a normal approximation to the binomial distribu

tion, i.e. B(n, p) N(np, npq)

Normal Distribution and Gauss Curve

NSS Mathematics Curriculum

Abraham de Moivre

• Abraham de Moivre• 1667-1754• French mathematician• a friend of Newton, Halley, Stirling

Motive

• Miscellanea Analytica (1730) ("On the Binomial a+b raised to high powers") began by quoting extensively from that portion of Ars Conjectandi where Bernoulli had first come to grips with the problem of specifying the number of experiments needed to determine the actual ratio of cases, within a given approximation.

• The mathematical treatment was his own.

• He began from the simpliest case, the symmetrical binomial, i.e. p = ½.

Mathematical Tools

• Newton

• Walli

• Bernoulli

3 5 7

11 3

2 4

56

1ln( ) 2( )

1 3 5 7

2 2 4 4 6 6

2 1 3 3 5 5 7

( 1)( 2)

1 2 2 3 4( 1)( 2)( 3)( 4)

2 3 4 5 6

c cc c c

c

x x x xx

x

n n c c c cn B n B n

c cc c c c c

B n

Theorem 1

/2 2

2 2

nnn

C

n

[to compare the middle term with the sum of all terms][when n = 2m]

Proof

mm

mm

m

m

m

m

m

mm

m

m

m

m

mm

n

nn

mm

mm

m

m

m

m

m

m

m

mm

m

m

m

m

m

m

mm

m

CC

1

1

3

3

2

2

1

121

21

2

2

2

22/

1

1

1

1

1

1

1

12ln

)1(

)1(

3

3

2

2

1

12ln

2

1

12

3

3

2

2

1

1

2

1ln

!!2

)!2(ln

2ln

2ln

Proof

7151311

753

753

753

1

1

3

3

2

2

1

121

)(7

1)(

5

1)(

3

12

)3

(7

1)

3(

5

1)

3(

3

132

)2

(7

1)

2(

5

1)

2(

3

122

)1

(7

1)

1(

5

1)

1(

3

112

2ln)21(

1

1ln

1

1ln

1

1ln

1

1ln2ln

mm

mm

mm

mm

mm

mm

m

m

m

m

m

mm

mmmm

mmmm

mmmm

m

Proof

6543212ln)2ln(2)12ln()

2

12(

)1(2

7)1(

4

35)1(

2

7)1(

2

1)1(

8

1)(2

)1(2

5)1(

2

5)1(

2

1)1(

6

1)(2

)1(2

3)1(

2

1)1(

4

1)(2

)1(2

1)1(

2

1 )

1( 2

2ln)21(

642

26

44

62

78

71

24

42

56

51

22

34

31

2

7

5

3

BBBmmmm

mBmBmBmm

mBmBmm

mBmm

mmm

m

m

m

m

Proof

(Stirling) 2ln1680

1

1260

1

360

1

12

11ln where

2

2

7976.0

2

7739.012ln2

1

2ln

7739.0654321

2ln Also,

4

11)

2

11ln(2))2ln(2)12(ln(2 that Note

2

2

2

2

642

B

nBm

C

mC

BBBmm

mmmmm

m

mm

m

mm

Another Try

22 2

2

2 2

/2

2 2 4 4 6 6 (2 )(2 ) 2 ( !) 1lim lim

2 1 3 3 5 5 7 (2 1)(2 1) (2 )! 2 1

(2 )! 2 1

2 2 ! ! (2 1)

2

2 2

m

m m

mmm m

nnn

m m m

m m m m

C m

m m m m

C

n

Theorem 2

2 2

2

2ln

mmmm

C

C n

[to compare the middle term with the term distant from it by an interval l]

Proof21 3 11 2

2 3 11 2

1 12 2

1 12 2

1 1 1 1ln ln

1 1 1 1

( ) ln( 1) ( ) ln( 1) 2 ln ln

1 1( ) ln(1 ) ( ) ln(1 ) ln(1 )

( ) ln(1 ) ( ) ln(1 )

(

m mm m m mm

m mm m m mm

C m

C m

mm m m m m m

m

m mm m m

m mm m

2 2

2 2

2 2

)( ) ( )( )2 2

2

m mm m m m

m n

Corollary

2 2

2 22 2

2

2 22 2

2 2

2 112 4

2 2/ 2

0 0

2

2 2 2

Note that ,

2 2 4

2 2 2

n

n

m mm m n nm m

xn xn n

d

C Ce e

n

p npq n

P e e dx e dxn n

De Moirve : 2 inflectional points nn

2

1

2

Areas Within 1, 2, 3 Standard Deviations

2

3

2 2 4 6 8 10 6

2 3 4 5 6

3 5 7 9 11

2 3 4 5

3( ) ( ) 3 ( ) 3 ( 2 ) ( 3 )

8

2 4 8 16 32 641

2 6 24 120 720integrate between 0 and to get

2 4 8 16 32

3 5 2 7 6 9 24 11 120

a h

a

n

hf x dx f a f a h f a h f a h

en n n n n n

n n n n n

Areas Within 1, 2, 3 Standard Deviations

De Moivre Exact

1 0.682 688 0.682 689

2 0.954 28 0.954 50

3 0.998 74 0.997 30

Significance of √n

• De Moivre introduced the term Modulus for the unit √n

• accuracy increases as √n

• Bernoulli had announced in Ars Conjectandi, even " the most stupid of men... is convinced that the more observation s have been made, the less danger there is of wandering from one's aim"

• De Moivre: more finely tuned analytical technique

Bernoulli's Failure

• Bernoulli's upper bound : 25550

• (De Moivre: 6498)

• Moral certainty: a high standard of certainty

• The entire population of Basel was smaller than 25550

• Flamsteed's (English astronomer) 1725 catalogue listed only 3000 stars

De Moivre's Success

• Bernoulli: To study the behaviour of the ratio of success (when n tends to infinity)

• De Moivre: To study the distribution of the occurrence of the favorable outcomes

De Moivre's Deficiency

• De Moirve's result failed to provided usable answers to the inference questions being asked at that time.

• Given known datum a(success) and b(failure), his formula can evaluate the chance, but if a and b are unknown, then it cannot give the chance that the unknown a/(a+b) would fall within the same specified distance of a given observed ratio.

De Moivre's Version of Stirling Theorem

1 1 1 11

1 1 1 11

2

1 2 3 11 1 1 1

( 1)!

1 2 3 1ln ln 1 1 1 1

( 1)!

(Newton's expansion and Bernoulli's summation formula)

1= 1 ln (

2 1 2

m

m

m m

m m m m m

m m

m m m m m

Bm m

1 3 54 61 ) (1 ) (1 )3 4 5 6

B Bm m m

After the publication of the first Miscellanea Analytica and then Stirling's book, de Moirve felt the need for rewriting and reorganizing his discussion on approximating the binomial, ...

De Moivre's Version of Stirling Theorem

12 3 5 7

1 1 1 1ln ! ( ) ln

12 360 1260 1680

1 1 1 1where exp(1 ) 2

12 360 1260 1680

x x x x Bx x x x

B

3

1 1! 2 exp( )

12 360xx x x x

x x

Reference

• Anders Hald, A History of Probability and Statistics and Their Applications before 1750 (p.468-495), John Wiley & Sons, 2003

• Stephen M. Stigler, The History of Statistics -The Measurement of Uncertainty before 1900 (p.62-98), Belknap Press, 1986

• A. De Moirve, The Doctrine of Chances (p.235-243) 2nd ed., London, 1738

• 徐傳勝 , 張梅東 , 正態分佈兩發現過程的數學文化比較 ,純粹數學與應用數學 CSCD 2007年第 23卷第 1期 137-144頁