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
random 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
distribution, 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
1 x
x3 x5 x7
ln(
)  2( x    
1 x
3 5
7

2
• Bernoulli

)
224466
1 3  3  5  5  7 
c 1
c
n
n
c
c( c  1)(c  2)
c
c 1
c 3
n



B
n

B
n

 c 1 c 2 2
4
2  3 4
c( c  1)(c  2)( c  3)( c  4)
B6n c 5 
2  3 4 5 6
Theorem 1
Cnn/2
2

n
2
2 n
[to compare the middle
term with the sum of all
terms]
[when n = 2m]
Proof
ln
Cnn / 2
2
 ln
n
 ln
2m
Cm
2 2m
( 2m )!
2 2m m! m!
2m  1 2m 
 1 m 1 m  2 m  3





 ln 

2m m  1 m  2 m  3
m
1

2
 1 2m m  1 m  2 m  3
m  ( m  1) 



 ln 2

)
1

m
(

m
3

m
2

m
1

m


 1 2m 1 
 ln 2
1

m 1 
3
2

1

1

1
m
m 
m
m
m 1 
3
2
1

1

1

1
m 
m
m
m
1
Proof
1 2 m
 ln 2
 ln
1  1m
1 m
 (1  2m ) ln 2 
1
 ln
1 2m
1 2
m
 ln
1  3m
1 m
3
   ln
1 1 1 3 1 1 5 1 1 7

 2   ( )  ( )  ( )  
5 m
7 m
m 3 m

1 2
1 2
2 1 2

 2   ( )3  ( )5  ( )7  
5 m
7 m
m 3 m

3 1 3 3 1 3 5 1 3 7

 2   ( )  ( )  ( )  
5 m
7 m
m 3 m


 m 1 1 m 1 3 1 m 1 5 1 m 1 7

 2  m  ( m )  ( m )  ( m )  
3
5
7


1  m 1 m
1  m 1 m
Proof
 (1  2m ) ln 2 
1 1
1

 2 ( )  ( m  1) 2  ( m  1)
m 2
2

1
3
1

 2( 1 3 )  ( m  1) 4  ( m  1) 3  B2 ( m  1) 2 
3m  4
2
2

1
5
5
1

 2( 1 5 )  ( m  1) 6  ( m  1) 5  B2 ( m  1) 4  B4 ( m  1) 2 
5m  6
2
2
2

1

8 1
7 7
6 35
4 7
1
 2( 7 )  ( m  1)  ( m  1)  B2 ( m  1) 
B4 ( m  1)  B6 ( m  1) 2 
7m  8
2
2
4
2


B
B
B
1
 ( 2m  ) ln(2m  1)  2m ln(2m )  ln 2  2  4  6  
2
1 2 3 4 5  6
Proof
1
1
Note that 2m(ln(2m  1)  2m ln(2m))  2m ln(1  )  1 

2m
4m
B6
B2
B4
Also,
ln 2 


   0.7739
1 2 3 4 5  6
Cm2m
1
ln
 ln 2m  1  0.7739
2 2m 2
2m
Cm
0.7976 2


2m B n
2 2m
1
1
1
1
where ln B  1  


   ln 2 (Stirling)
12 360 1260 1680
Another Try

2
 2 ( m!) 
2  2  4  4  6  6 (2m)(2m)
1
 lim
 lim 


2 m 1  3  3  5  5  7 (2m  1)(2m  1) m  (2m)!  2m  1
Cm2 m
(2m)!
2
1



22 m 22 m m ! m !
 (2m  1)
m
Cnn/2
2

n
2
2 n
2m
2
Theorem 2
Cm2 m
2
ln 2 m 
Cm
n
2
[to compare the middle term
with the term distant from it by an interval l]
Proof
Cm21m
 m  1  1m 1  2 m 1  3 m
ln 2 m   ln 




3
1
2
Cm
 m 1 m 1 m 1 m
1  m 1 m 
1  m 1 m 

m
 ( m   ) ln( m   1)  ( m   ) ln( m   1)  2m ln m  ln
m
1
1
1
1
 ( m   2 ) ln(1 
)  ( m   2 ) ln(1 
)  ln(1  )
m
m
m
1
2
1
2
 ( m  ) ln(1 
 ( m  )(
m
 2 2


m
n
2

m
)  ( m  ) ln(1 
2
2m
2

m
)  ( m  )( 
)
m

2
2m
2

)
Corollary
2m
m
2m
C
2
2m
m
2m
C

2
e
2
n
Note that p 
n

2
Pd  
n

2
1
2
2
2
e
2 n
2
e

2 n
2
n
2
,  2  npq  14 n
2
n
2
2

2 n

0
e
2 x 2
n
4
dx 
2
n 1

n
De Moirve : 2 inflectional points
2 2

0
/ n
e
2 x 2
dx
Areas Within 1, 2, 3 Standard Deviations

a  3h
a
2
n
3h
f ( x )dx   f (a )  3 f (a  h )  3 f ( a  2h )  f ( a  3h) 
8
2
2 2 4 4 8 6 16 8 32 10
64 6
e
 1
 2 3



4
5
6
n
2n 6n 24n 120n 720n
integrate between 0 and to get
2 3
4 5
8 7
16 9
32 11






2
3
4
5
3n 5  2n 7  6n 9  24n 11  120n
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
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, ...
m 1
1
1
m
1 
2 
3

 1   1   1  
( m  1)!  m   m   m 
m 1
1
1
1
 m 1
1 

m


1
1
1
m
1 
2 
3

 m 1
ln
 ln  1    1    1  
1 

( m  1)!
m
m
m
m

 
 



 (Newton's expansion and Bernoulli's summation formula)
1
B2
B4
B6
1
3
=m  1  ln m 
(1  m ) 
(1  m ) 
(1  m 5 ) 
2
1 2
3 4
56
De Moivre's Version of Stirling Theorem
1
1
1
1
ln x !  ( x  ) ln x  x  B 




3
5
7
12 x 360 x 1260 x 1680 x
1
2
1
1
1
1
where B  exp(1  



12 360 1260 1680
x !  2 x  x x  exp(  x 
)  2
1
1


3
12 x 360 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137144頁