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Random variable

About: Random variable is a research topic. Over the lifetime, 29109 publications have been published within this topic receiving 674687 citations. The topic is also known as: random quantity & aleatory variable.


Papers
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Journal ArticleDOI
TL;DR: In this article, the authors discuss the problem of estimating the sampling distribution of a pre-specified random variable R(X, F) on the basis of the observed data x.
Abstract: We discuss the following problem given a random sample X = (X 1, X 2,…, X n) from an unknown probability distribution F, estimate the sampling distribution of some prespecified random variable R(X, F), on the basis of the observed data x. (Standard jackknife theory gives an approximate mean and variance in the case R(X, F) = \(\theta \left( {\hat F} \right) - \theta \left( F \right)\), θ some parameter of interest.) A general method, called the “bootstrap”, is introduced, and shown to work satisfactorily on a variety of estimation problems. The jackknife is shown to be a linear approximation method for the bootstrap. The exposition proceeds by a series of examples: variance of the sample median, error rates in a linear discriminant analysis, ratio estimation, estimating regression parameters, etc.

14,483 citations

Book
01 Jan 2002
TL;DR: In this paper, the meaning of probability and random variables are discussed, as well as the axioms of probability, and the concept of a random variable and repeated trials are discussed.
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

12,407 citations

Journal ArticleDOI
TL;DR: In this paper, the authors show that the limit distribution is normal if n, n$ go to infinity in any arbitrary manner, where n = m = 8 and n = n = 8.
Abstract: Let $x$ and $y$ be two random variables with continuous cumulative distribution functions $f$ and $g$. A statistic $U$ depending on the relative ranks of the $x$'s and $y$'s is proposed for testing the hypothesis $f = g$. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis $f = g$ the probability of obtaining a given $U$ in a sample of $n x's$ and $m y's$ is the solution of a certain recurrence relation involving $n$ and $m$. Using this recurrence relation tables have been computed giving the probability of $U$ for samples up to $n = m = 8$. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if $m, n$ go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives $f(x) > g(x)$ for every $x$.

11,055 citations

Journal ArticleDOI
TL;DR: In this paper, the problem of the estimation of a probability density function and of determining the mode of the probability function is discussed. Only estimates which are consistent and asymptotically normal are constructed.
Abstract: : Given a sequence of independent identically distributed random variables with a common probability density function, the problem of the estimation of a probability density function and of determining the mode of a probability function are discussed. Only estimates which are consistent and asymptotically normal are constructed. (Author)

10,114 citations

Journal ArticleDOI
TL;DR: The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable.

8,918 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023379
2022786
20211,294
20201,435
20191,454
20181,425