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: 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 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, -m+1,…,0,…,m-1,m with prob. p-m ,…,pm. Let Sn be the sum of the n possible errors.

CLT for the Binomial Distribution

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

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 “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 “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 “(…) the correct way of relating past events to the probability of causes and of future events (…)” CAUSE EVENT FUTURE EVENT

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