Advanced Level Mathematics: Statistics 1 by Steve Dobbs

By Steve Dobbs

Written to check the contents of the Cambridge syllabus. information 1 corresponds to unit S1. It covers illustration of information, diversifications and mixtures, likelihood, discrete random variables and the traditional distribution.

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60) 2. For all r E I the set G ( r ) = {x;g(~, r) = O} is a compact Cl-manifold in 1Rn and for all x G G(r) always ~Txg(x, r) 5s o. (An orientation for G(v) is given by the normal field nr = IVxg(x, r)[-1V~eg(x, r) ). 61) g(x,~)<0 exists for all r G I and the derivative of this integral with respect to r is given by F'(T)= j f~(x, T) dx - j g(m,~)

32) [] From this result some special cases can be derived. C o r o l l a r y 37 Let f and h be continuous functions on a finite interval [c~,fl]. 33) b) If the global maximum occurs only at ~, h(~) # 0 and f ( x ) is near c~ continuously differentiable with f ' ( c 0 < O, then 1 I(A) ~ h(o~)exp(Af(cr))Alf,(oOi , A ~ oo. 34) c) If the global maximum occurs only at ~, h(o0 # 0 and f ( x ) is twice continuously differentiable near ct with f'(c~) = 0 and f " ( ~ ) < O, then Z(,~) ~ h(c~)exp(,~f(c~)) PROOF: 2,Xlf,,(cO [ , ,~ ~ ~ .

20) Gt is finite. 21) o, with A > 1 are all finite and have the asymptotic approximation 8 I(~) ,~ b F r r r A--~ exp(~f(c~)), a ---+ oo. 22) PP~OOF: For simplicity we assume that f ( a ) = 0. To show that all integrals I(A) are finite, we get an upper bound for them II(~)1 ~ /Ih(x){exp(~f(x)) dx t~ P <_ f [h(x)lexp(Af(x)) dx + f Ih(x)lexp(Af(x)) dx. 23) The first integral is finite. 24) o'd-e = Ih(x)l exp (f(x)) dx = O(exp(-)~6)). exp(-6(,~ - 1)) f c~+e But since due to assumption 3) the last integral is finite, we find that I I ( ~ ) 1 < for all ,~ > 0.

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