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Advanced Level Mathematics: Statistics 1 by Steve Dobbs

By Steve Dobbs

Written to compare the contents of the Cambridge syllabus. statistics 1 corresponds to unit S1. It covers illustration of information, variations and mixtures, likelihood, discrete random variables and the conventional distribution.

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S. (1974) . Strong convexity of the pressure for lattice [DN) systems of classical statistical physics. Teoret , Mat . Fiz. 20, 223-234. [DS1) Dobrushin, R . L . and Shlosman, S. B . (1987). Completely analytic interactions. Constructi ve description. Journal of Stat istical Phys ics 46, 983-1014. [DS2) Dobrushin, R. L. and Shlosman, S. B . (1992) . Large deviation behavior of statistical mechanics models in the multiphaseregime. Proc eedings of the Xth Congress on Mathematical Physics, Leip zig 1991 (K .

I. (1977) . Independent and Stationary Sequences of Ran[IL] dom Variables. Walters-Noordhoff, Groningen. Isakov, S . N. (1984) . Nonanalytic features of the first order phase transition in the Ising (1) model. Communications in Mathemati cal Physics 95 , 427-443. Khinchin, A. I. (1929) . Ub er einen n euen grenzwertsatz der wahrscheinlichkeitsreclumng. [Kh] Math ematische Annalen 101, 745 -752 . Kotecky, R . and Preiss, D . (1986). Cluster expansion for abstract polymer models. Com[KP) munications in Mathematical Physics 103, 491-498.

The total deviation of SN is caused by deviations of the sizes of separate droplets, which ar e almost independent , and so its probability behaves in the same way as in the classical case of independent variables. If we increase the number of minus-particles (the case bN < Et), the same situation holds, provided the order of the number of additional minus-particles is smaller than IVNI(II-1)/II . But after we pass this threshold, the microscopic droplets can no longer absorb all the additional minus-particles, so a large minus-droplet of a macroscopic size is formed, which contains the main part of thes e particles.

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