Topic

# Independent and identically distributed random variables

About: Independent and identically distributed random variables is a research topic. Over the lifetime, 8148 publications have been published within this topic receiving 214402 citations. The topic is also known as: IID & iid.

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Book

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01 Jan 1965
TL;DR: This chapter discusses the concept of a Random Variable, the meaning of Probability, and the axioms of probability in terms of Markov Chains and Queueing Theory.
Abstract: Part 1 Probability and Random Variables 1 The Meaning of Probability 2 The Axioms of Probability 3 Repeated Trials 4 The Concept of a Random Variable 5 Functions of One Random Variable 6 Two Random Variables 7 Sequences of Random Variables 8 Statistics Part 2 Stochastic Processes 9 General Concepts 10 Random Walk and Other Applications 11 Spectral Representation 12 Spectral Estimation 13 Mean Square Estimation 14 Entropy 15 Markov Chains 16 Markov Processes and Queueing Theory

13,886 citations

Book ChapterDOI

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TL;DR: In this article, upper bounds for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt are derived for certain sums of dependent random variables such as U statistics.
Abstract: Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S is bounded or bounded above. The bounds for Pr {S – ES ≥ nt} depend only on the endpoints of the ranges of the summands and the mean, or the mean and the variance of S. These results are then used to obtain analogous inequalities for certain sums of dependent random variables such as U statistics and the sum of a random sample without replacement from a finite population.

8,475 citations

Journal ArticleDOI

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4,025 citations

Journal ArticleDOI

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TL;DR: In this paper, the authors describe six different statistical approaches to infer correlates of species distributions, for both presence/absence (binary response) and species abundance data (poisson or normally distributed response), while accounting for spatial autocorrelation in model residuals: autocovariate regression; spatial eigenvector mapping; generalised least squares; (conditional and simultaneous) autoregressive models and generalised estimating equations.
Abstract: Species distributional or trait data based on range map (extent-of-occurrence) or atlas survey data often display spatial autocorrelation, i.e. locations close to each other exhibit more similar values than those further apart. If this pattern remains present in the residuals of a statistical model based on such data, one of the key assumptions of standard statistical analyses, that residuals are independent and identically distributed (i.i.d), is violated. The violation of the assumption of i.i.d. residuals may bias parameter estimates and can increase type I error rates (falsely rejecting the null hypothesis of no effect). While this is increasingly recognised by researchers analysing species distribution data, there is, to our knowledge, no comprehensive overview of the many available spatial statistical methods to take spatial autocorrelation into account in tests of statistical significance. Here, we describe six different statistical approaches to infer correlates of species’ distributions, for both presence/absence (binary response) and species abundance data (poisson or normally distributed response), while accounting for spatial autocorrelation in model residuals: autocovariate regression; spatial eigenvector mapping; generalised least squares; (conditional and simultaneous) autoregressive models and generalised estimating equations. A comprehensive comparison of the relative merits of these methods is beyond the scope of this paper. To demonstrate each method’s implementation, however, we undertook preliminary tests based on simulated data. These preliminary tests verified that most of the spatial modeling techniques we examined showed good type I error control and precise parameter estimates, at least when confronted with simplistic simulated data containing

2,535 citations

Book

[...]

01 Jan 1977
TL;DR: In this paper, the optimal linear non-stationary filtering, interpolation and extrapolation of Partially Observable Random Processes with a Countable Number of States (POMOS) was studied.
Abstract: 1. Essentials of Probability Theory and Mathematical Statistics.- 2. Martingales and Related Processes: Discrete Time.- 3. Martingales and Related Processes: Continuous Time.- 4. The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations.- 5. Square Integrable Martingales and Structure of the Functionals on a Wiener Process.- 6. Nonnegative Supermartingales and Martingales, and the Girsanov Theorem.- 7. Absolute Continuity of Measures corresponding to the Ito Processes and Processes of the Diffusion Type.- 8. General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes.- 9. Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States.- 10. Optimal Linear Nonstationary Filtering.

2,481 citations

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##### Performance
###### Metrics
No. of papers in the topic in previous years
YearPapers
202390
2022184
2021308
2020280
2019315
2018283