Random variable

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  • This chapter is of an introductory nature, its purpose being to indicate some concepts and results from the theory of probability which are used in later chapters . Most of these are contained in Chapters 1-9 of Gnedenko [47], and will therefore be cited without proof. The first section is somewhat isolated, and contains a series of results from the foundations of the theory of probability. A detailed account may be found in [76], or in Chapter I of [31] . Some of these will not be needed in the first part of the book, in which attention is confined to independent random variables ....

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  • Tham khảo sách 'probability examples c-3 random variables ii', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả

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  • Tham khảo sách 'probability examples c-4 random variables iii', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả

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  • Topic 12 - Forward contracts and hedges, simulated correlated random variables. After completing this unit, you should be able to: Compute no-arbitrage forward prices for equities, currencies, and commodities; compute the payoffs of forward contracts; construct forward hedges using beta and correlation; simulate correlated random variables.

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  • Let us consider the composed random variable η = k=1 ξk , where ξ1 , ξ2 , ... are independent identically distributed random variables and ν is a positive value random, independent of all ξk . In [1] and [2], we gave some the stabilities of the distribution function of η in the following sense: the small changes in the distribution function of ξ k only lead to the small changes in the distribution function of η. In the paper, we investigate the distribution function of η when we have the small changes of the distribution of ν. ...

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  • This book is devoted to the theory of probabilistic information measures and their application to coding theorems for information sources and noisy chan- nels. The eventual goal is a general development of Shannon's mathematical theory of communication, but much of the space is devoted to the tools and methods required to prove the Shannon coding theorems. These tools form an area common to ergodic theory and information theory and comprise several quantitative notions of the information in random variables, random processes, and dynamical systems....

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  • Chapter 3 REFINEMENTS OF THE LIMIT THEOREMS FOR NORMAL CONVERGENCE § 1 . Introduction In this chapter we consider a sequence X 1 , X2 , . . . of independent, identically distributed random variables belonging to the domain of attraction of the normal law. As shown in § 2 .6, the X; necessarily have a finite variance a 2 .

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  • Chapter 7 RICHTER'S LOCAL THEOREMS AND BERNSTEIN'S INEQUALITY 1 . Statement of the theorems The theorems of this chapter do not have a collective character, and are related to Theorem 6 .1.1 . We shall consider a sequence of independent, identically distributed random variables XX

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  • Chapter 12 WIDE MONOMIAL ZONES OF INTEGRAL NORMAL ATTRACTION 1 . Formulation In this chapter, as before, we study the independent, identixally distributed random variables X1, X2, . . . with E (Xi) = 0, V (Xl) = 1 . We shall study the zone [0, n"] where a 6 ; we recall that this is said to be a zone of normal attraction if,

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  • Chapter 4 LOCAL LIMIT THEOREMS § 1. Formulation of the problem Suppose that the independent, identically distributed random variables X1 , X2 ,. . . . have a lattice distribution with interval h, so that the sum Zn = X1 + X2 + . . . + X„ takes values in the arithmetic progression {na + kh ; k = 0, ± 1, . . . } .

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  • Chapter 5 LIMIT THEOREMS IN Lp SPACES § 1 . Statement of the problem Consider the sequence X1 , X2 , . . . of independent random variables with the same distribution F. If F belongs to the domain of attraction of a stable law G« with exponent a, then the distribution functions Fn of the normalised sums Zn = (X1 + X2+ . . . + Xn - An)/Bn satisfy lim Fn (x) = G a (x)

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  • Chapter 6 LIMIT THEOREMS FOR LARGE DEVIATIONS § 1 . Introduction and examples In this and succeeding chapters we shall examine the simplest problems in the theory of large deviations . Let X1 , X2 ,. . . be independent, identically distributed random variables, with E(X1) = 0

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  • Chapter 9 MONOMIAL ZONES OF LOCAL NORMAL A 11 RACTION 1 . Zones of normal attraction In this chapter it will be assumed that the independent random variables X; satisfy E(X;)=0, V(XX)=62 0, and that Sn = X1 + X2+ .. . +X,, Z n = Sn/ an-1 .

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  • Chapter 14 INTEGRAL THEOREMS HOLDING ON THE WHOLE LINE 1 . Formulation In the preceding chapters we have studied theorems of a collective type concerning large deviations in zones of the form [0, 0 (n)] and [ - (n), 0], where 0 (n) = o (n 2) . The role of the linear functionals a j, bj was played by moments of the random variables XX .

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  • Chapter 15 APPROXIMATION OF DISTRIBUTIONS OF SUMS OF INDEPENDENT COMPONENTS BY INFINITELY DIVISIBLE DISTRIBUTIONS 1 . Statement of the problem We here consider the general problem of the limiting behaviour of the distribution function F§ (x) of the sum (15.1 .1) of independent random variables with the same distribution F

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  • Chapter 19 EXAMPLES AND ADDENDA The separate sections of this chapter are not related to one another except in so far as they illustrate or extend the results of Chapter 18 . © 1 . The central limit theorem for homogeneous Markov chains Consider a homogeneous Markov chain with a finite number of states (labelled 1, 2, . . ., k) and transition matrix P = (p i ;) (see, for instance, Chapter III of [47] ) . If Xn is the state of the system at time n, we have the sequence of random variables X1 , X2 , . . ., Xn...

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  • Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 279754, 13 pages doi:10.1155/2011/279754 Research Article On Strong Law of Large Numbers for Dependent Random Variables Zhongzhi Wang Faculty of Mathematics and Physics, Anhui University of Technology, Ma’anshan 243002, China Correspondence should be addressed to Zhongzhi Wang, wzz30@ahut.edu.cn Received 16 December 2010; Accepted 3 March 2011 Academic Editor: Vijay Gupta Copyright q 2011 Zhongzhi Wang.

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  • Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables

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  • Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Strong Limit Theorem for Weighted Sums of Sequences of Negatively Dependent Random Variables

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  • (BQ) Part 1 book "Introductory statistics" has contents: The nature of statistics, organizing data, descriptive measures, probability concepts, discrete random variables, the normal distribution, the sampling distribution of the sample mean, confidence intervals for one population mean, hypothesis tests for one population mean.

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