By V. M. Kalinin
ISBN-10: 1468482114
ISBN-13: 9781468482119
ISBN-10: 1468482130
ISBN-13: 9781468482133
Read or Download Investigations in Classical Problems of Probability Theory and Mathematical Statistics: Part I PDF
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Extra info for Investigations in Classical Problems of Probability Theory and Mathematical Statistics: Part I
Sample text
91 D l_)Vti \ ~v '" ~ t\ lItlHtV)(t) dt . 31) e Rem ark 2. Theorem 2 can be expressed in the following two ways: l-t + III 1 \ \ e 91 ~ l\tIHtV)lt)dt,. +t)dt e ~ t o + ~! ('l:) e and 91 satisfy the fol- and its first V derivatives. 33) can be derived from Theorem 6 with the help of obvious transformations and the properties of the Bernoulli polynomials. It is sometimes necessary to sum a function over integral pOints contained in some interval whose end points mayor may not be integers. The arbitrariness in the choice of e and at can be used.
T~) can be expanded in a Taylor's series converging to h~) in the interval Q+ h, t ~ -") "'r. ,,0.. + h,~, with ~ =I.... , 1-\,. Here, we have h, =- t, ~Q. 53) corresponds to the trapezoidal formula. 54) is a generalization of the concept of Riemann sums when e ~ [0, n . The other direction is the derivation of expansion formulas for functions. -Q. ) H J. 55) which with e = 0 and e = -I coincides with the Taylor's series. ), we can obtain a variety of expansions of functions in special series. 55) was the first to hand.
2). 9) for ~=~,2, ... and JA--O'~, ... ~ JA- .. ~ - '2. Let us separate the first term (with J .. p' (9) = (_tl e,,II-+1(9) • (f'~)! Theorem 4 has thus been proved. Rem ark 1. If the function f ('X,) and its first Vderivatives are continuous on the whole of the real axis, then there are no restrictions on the choice of e. This is all the more so when Hl) is an entire function. Rem ark 2. Theorem 4 is very curbersome and we will only use its limiting cases. 5)]. 1) vanishes. 5) when V,. ~ t "'-8 ttL) - \ e 8·1 e ~(~)d'X, +\ 1,9-i -t)t~' (i.
Investigations in Classical Problems of Probability Theory and Mathematical Statistics: Part I by V. M. Kalinin
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