A history of the mathematical theory of probability : from by Todhunter, I. (Isaac)

By Todhunter, I. (Isaac)

The beneficial reception which has been granted to my heritage of the Calculus of adaptations through the 19th Century has inspired me to adopt one other paintings of a similar type. the topic to which I now invite recognition has excessive claims to attention as a result of the delicate difficulties which it includes, the dear contributions to research which it has produced, its very important sensible purposes, and the eminence of these who've cultivated it.

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A history of the mathematical theory of probability : from the time of Pascal to that of Laplace

The beneficial reception which has been granted to my heritage of the Calculus of diversifications through the 19th Century has inspired me to adopt one other paintings of an analogous sort. the topic to which I now invite awareness has excessive claims to attention because of the delicate difficulties which it comprises, the precious contributions to research which it has produced, its vital functional functions, and the eminence of these who've cultivated it.

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1 Henee, (5) holds if we have ( " 1: Xf-l ;= )II(P-I) (" + 1: yf-l )1/(1)-1) (7) i = I I ~ [ "JIM-I) ,1: (x i + Yi)P-l ,= I . ° This, however, is MINKOWSKI'S inequality, valid for 1 < The inequality sign in (1) is reversed for ~ p ~ 1. P~ 2. § 24. An Inequality of Dresher An extension of BECKENBACH'S inequality was obtained by [1] by means of moment-space teehniques: Theorem 10. p . DRESHER ( 1) This result ean be derived through quasi linearization, as in § 23. DANSKIN [2], who employed a eombination of the Hölder and Minkowski inequalities.

CHASSAN [6J. § 45. Refinements of the Cauchy-Buniakowsky-Schwarz Inequalities Having established the nonnegativity of the functional I (u, v) = (f u dt) (f 2 V2 dt) - (f uv dtr (1) we naturally are interested in obtaining a more precise lower bound than zero. We can do this whenever the functions or functionals under consideration are quadratic in the following fashion. Reverting to inner products, consider the function J (u, v) = (u, u) (v, v) - (u, V)2, (2) assumed nonnegative for all u and v.

Similarly, for a strictly f(u) = min [f(v) + (u-v)f'(v)J. v (1) (2) The general resuIt is Theorem 11. Let f(x) = f (Xl' x 2 , ••• , x n ) be a strictly convex function of x for all x; then f(x) = max [f(y) + (x - y, ep (y))] , (3) ,. 30 1. The Fundamental Inequalities and Related Matters where 1>(Y) = (of/oYl' of/oY2' ... ), the gradient of f(y). The unique maximum occurs at Y = x. This type of quasi linearization has been extensively used by BELLMAN [1], [2] and KALABA [3] in eonneetion with the analytic and eomputational treatment of nonlinear funetional equations.

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