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De Moivre and Normal Distribution

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.

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De Moivre and Normal Distribution

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  1. De Moivre and Normal Distribution The First Central Limit Theorem

  2. 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)

  3. Normal Distribution and Gauss Curve

  4. NSS Mathematics Curriculum

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

  6. 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.

  7. Mathematical Tools • Newton • Walli • Bernoulli

  8. Theorem 1 [to compare the middle term with the sum of all terms] [when n = 2m]

  9. Proof

  10. Proof

  11. Proof

  12. Proof

  13. Another Try

  14. Theorem 2 [to compare the middle term with the term distant from it by an interval l]

  15. Proof

  16. Corollary De Moirve : 2 inflectional points

  17. Areas Within 1, 2, 3 Standard Deviations

  18. Areas Within 1, 2, 3 Standard Deviations

  19. 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

  20. 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

  21. 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

  22. 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.

  23. 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, ...

  24. De Moivre's Version of Stirling Theorem

  25. 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

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