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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Algebra of random variables ： ウィキペディア英語版 Algebra of random variables The algebra of random variables provides rules for the symbolic manipulation of random variables, while avoiding delving too deeply into the mathematically sophisticated ideas of probability theory. Its symbolism allows the treatment of sums, products, ratios and general functions of random variables, as well as dealing with operations such as finding the probability distributions and the expectations, variances and covariances of such combinations. ==Mathematical basis== In the algebraic axiomatization of probability theory, the primary concept is not that of probability of an event, but rather that of a random variable. Probability distributions are determined by assigning an expectation to each random variable. The measurable space and the probability measure arise from the random variables and expectations by means of well-known representation theorems of analysis. One of the important features of the algebraic approach is that apparently infinite-dimensional probability distributions are not harder to formalize than finite-dimensional ones. Random variables are assumed to have the following properties: # complex constants are random variables; # the sum of two random variables is a random variable; # the product of two random variables is a random variable; # addition and multiplication of random variables are both commutative; and # there is a notion of conjugation of random variables, satisfying and for all random variables and coinciding with complex conjugation if is a constant. This means that random variables form complex commutative *-algebras. If then the random variable is called "real". An expectation on an algebra of random variables is a normalized, positive linear functional. What this means is that # where is a constant; # for all random variables ; # for all random variables and ; and # if is a constant. One may generalize this setup, allowing the algebra to be noncommutative. This leads to other areas of noncommutative probability such as quantum probability, random matrix theory, and free probability. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Algebra of random variables」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.