Khinchin's Theorem On The Factorization Of Distributions

	Khinchin's Theorem On The Factorization Of Distributions Khinchin's theorem on the factorization of distributions says that every probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ... ''P'' admits (in the convolution semi-group of probability distributions) a factorization :P = P_1 \otimes P_2 where ''P''1 is a probability distribution without any indecomposable factor and ''P''2 is a distribution that is either degenerate or is representable as the convolution of a finite or countable set of indecomposable distributions. The factorization is not unique, in general. The theorem was proved by A. Ya. Khinchin for distributions on the line, and later it became clear that it is valid for distributions on considerably more general groups. A broad class (seeI.Z. Ruzsa, G.J. Székely, "Algebraic probability theory" , Wil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Probability Distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Indecomposable Distribution In probability theory, an indecomposable distribution is a probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables: ''Z'' ≠ ''X'' + ''Y''. If it can be so expressed, it is decomposable: ''Z'' = ''X'' + ''Y''. If, further, it can be expressed as the distribution of the sum of two or more independent ''identically'' distributed random variables, then it is divisible: ''Z'' = ''X''1 + ''X''2. Examples Indecomposable * The simplest examples are Bernoulli-distributeds: if ::X = \begin 1 & \text p, \\ 0 & \text 1-p, \end :then the probability distribution of ''X'' is indecomposable. :Proof: Given non-constant distributions ''U'' and ''V,'' so that ''U'' assumes at least two values ''a'', ''b'' and ''V'' assumes two values ''c'', ''d,'' with ''a'' < ''b'' and ''c'' < ''d'', then ''U'' + ''V'' ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]