Laplace, Pierre Simon de (1749-1827)

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Laplace, Pierre Simon de (1749-1827)

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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

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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.

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Biography (Contd)

• 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

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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/v2)exp(-x2/2), came with Thomas Simpson.

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History Of CLT (Contd)

• 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.

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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.vs following a triangular
  distribution between 0 and 90 degrees
• Problems
• 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 !!

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First Version of the CLT

• Laplace introduced the m.g.f,
  which is known as Laplace Transform of
  f
• He then introduced the Characteristic function
• If is a sample of i.i.d.
  obs.,
• Assume that we have a discrete r.v. x, that takes
  on the values
• m, -m1,,0,,m-1,m with prob. p-m ,,pm.
• Let Sn be the sum of the n possible errors.

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CLT for the Binomial Distribution
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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.
• Where

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Laplace Bayesian ideasOverview
Philosophical essay on probabilities by Laplace

• General principles on probability
• Expectation
• Analytical methods
• Applications

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4th 5th principles conditional marginal give
the joint
Here the question posed by some philosophers
concerning the influence of the past on the
probability of the future, presents itself
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6th principle discrete Bayes theorem
Fundamental principle of that branch of the
analysis of chance that consists of reasoning a
posteriori from events to causes
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7th principle probability of future based on
observations
() the correct way of relating past events to
the probability of causes and of future events
()
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References

• Laplace, Pierre Simon De. Philosophical essay on
  probabilities.
• Weatherburn, C.E. A First Course in Mathematical
  Statistics.