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Laplace, Pierre Simon de (1749-1827)
Agenda
Biography
History of The Central limit Theorem (CLT)
Derivation of the CLT
First Version of the CLT
CLT for the Binomial Distribution
Laplace and Bayesian ideas
4th and 5th Principles
6th Principle
7th Principle
References
Biography
Pierre Simon Laplace was born in Normandy on March 23,
1749, and died at Paris on March 5, 1827
French scientist, mathematician and astronomer; established
mathematically the stability of the Solar system and its origin without a divine intervention
Professor of mathematics in the École militaire of Paris at the
age of 19.
Biography (Cont’d)
Under Napoleon, Laplace was a member, then chancellor, of the
Senate, and received the Legion of Honour in 1805. Appointed
Minister of the Interior, in 1799. Removed from office by
Napoleon after six weeks only!!
Named a Marquis in 1817 after the restoration of the Bourbons
Main publications:
Mécanique céleste (1771, 1787)
Théorie analytique des probabilités 1812 – first edition dedicated
to Napoleon
History Of Central Limit Theorem
From De Moivre to Laplace
De Moivre investigated the limits of the binomial distribution
as the number of trials increases without bound and found
that the function exp(-x2) came up in connection with this
problem.
The formulation of the normal distribution, (1/√2)exp(-x2/2),
came with Thomas Simpson.
History Of CLT (Cont’d)
This idea was was expanded upon by the German
mathematician Carl Friedrich Gauss who then developed the
principle of least squares.
Independently, the French mathematicians Pierre Simon de
Laplace and Legendre also developed these ideas. It was with
Laplace's work that the first inklings of the Central Limit
Theorem appeared.
In France, the normal distribution is known as Laplacian
Distribution; while in Germany it is known as Gaussian.
Derivation of the CLT
Initial Work: Laplace was calculating the probability
distribution of the sum of meteor inclination angles. He
assumed that all the angles were r.v’s following a
triangular distribution between 0 and 90 degrees
Problems:
I.
II.
The deviation between the arithmetic mean which was
inflicted with observational errors, and the theoretical value
The exact calculation was not achievable due to the
considerable amount of celestrial bodies
Solution: Find an approximation !!
First Version of the CLT
tx
Laplace introduced the m.g.f, M x (t )  E[e ]which is
known as Laplace Transform of f
He then introduced the Characteristic function:
x (t )  M x (it )  E[eitx ]
If
x1, x2 ,..., xn is a sample of i.i.d. obs.,  (t )  (x (t ))
n
 xi
1
i 1
Assume that we have a discrete r.v. x, that takes on the values
–m, -m+1,…,0,…,m-1,m with prob. p-m ,…,pm.
Let Sn be the sum of the n possible errors.
n
1  ijt n
P( Sn  j ) 
 e x (t )
2 
m
x (t )   pk eik t
k  m
m
1  ijt
t2 m
2
n
P( Sn  j ) 
e
(
1

it
p
k

p
k

...)



k
k
k  m
2 
2 k  m
m
u x   pk k
k  m
m
   pk k 2  u x2
2
x
k  m
1 
1
2 2
P( Sn  j ) 
exp[

ijt

itu

n


x
x t  ...) dt
2 
2
1
  s2

P ( S n  nu x  j ) 
exp
2
2
2
n


2n x
x 
CLT for the Binomial Distribution
r  np  x
n! p np  x q nq  r
P
( np  x )! ( nq  x )!
n! ~
2ne  n n n
P 1
z  x/
N

x 

; N  1 
2npq
np


n
 2  pq
log N   z 2 / 2 pq
 ( z)  1

2
exp(  z
2
2 2
)
np  x 1 / 2

x 

1 
nq


nq  x 1 / 2
Final Note on The Proof of The CLT
It was Lyapunov's analysis that led to the modern characteristic
function approach to the Central Limit Theorem.
z (t )  z (t )
n
Where
x u
zn 
/ n
z ~ N (0,1)
Laplace & Bayesian ideas
Overview
“Philosophical essay on probabilities” by Laplace
• General principles on probability
• Expectation
• Analytical methods
• Applications
4th & 5th principles: conditional & marginal
give the joint
P( A, B)  P( A) P( B | A)
“Here the question posed by some philosophers
concerning the influence of the past on the probability of
the future, presents itself”
6th principle: “discrete” Bayes theorem
P(event | causei )
P(causei | event) 
 j P(event | cause j )
“Fundamental principle of that branch of the analysis of
chance that consists of reasoning a posteriori from events
to causes”
CAUSE
EVENT
7th principle: probability of future based on
observations
P( E future | Eobs )  i P( E future | causei ) P(causei | Eobs )
“(…) the correct way of relating past events to the
probability of causes and of future events (…)”
CAUSE
FUTURE
EVENT
EVENT
References
Laplace, Pierre Simon De. “Philosophical essay on
probabilities.
Weatherburn, C.E. “A First Course in Mathematical
Statistics”.