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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 Probability space ： ウィキペディア英語版 Probability space In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or "experiment") consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same. A probability space consists of three parts:〔Loève, Michel. Probability Theory, Vol 1. New York: D. Van Nostrand Company, 1955.〕〔Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press.〕 # A sample space, Ω, which is the set of all possible outcomes. # A set of events $\backslash scriptstyle\; \backslash mathcal$, where each event is a set containing zero or more outcomes. # The assignment of probabilities to the events; that is, a function ''P'' from events to probabilities. An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex ''events'' are used to characterize groups of outcomes. The collection of all such events is a ''σ-algebra'' $\backslash scriptstyle\; \backslash mathcal$. Finally, there is a need to specify each event's likelihood of happening. This is done using the ''probability measure'' function, ''P''. Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, ''ω'', from the sample space Ω. All the events in $\backslash scriptstyle\; \backslash mathcal$ that contain the selected outcome ''ω'' (recall that each event is a subset of Ω) are said to “have occurred”. The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function ''P''. The Russian mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. Nowadays alternative approaches for axiomatization of probability theory exist; see “Algebra of random variables”, for example. This article is concerned with the mathematics of manipulating probabilities. The article probability interpretations outlines several alternative views of what "probability" means and how it should be interpreted. In addition, there have been attempts to construct theories for quantities that are notionally similar to probabilities but do not obey all their rules; see, for example, Free probability, Fuzzy logic, Possibility theory, Negative probability and Quantum probability. ==Introduction== A probability space is a mathematical triplet $(\backslash Omega,\; \backslash mathcal,\; P)$ that presents a model for a particular class of real-world situations. As with other models, its author ultimately defines which elements $\backslash Omega$, $\backslash mathcal$, and $P$ will contain. * The sample space $\backslash Omega$ is a set of outcomes. An outcome is the result of a single execution of the model. Outcomes may be states of nature, possibilities, experimental results and the like. Every instance of the real-world situation (or run of the experiment) must produce exactly one outcome. If outcomes of different runs of an experiment differ in any way that matters, they are distinct outcomes. Which differences matter depends on the kind of analysis we want to do: This leads to different choices of sample space. * The σ-algebra $\backslash mathcal$ is a collection of all the events (not necessarily elementary) we would like to consider. Here, an "event" is a set of zero or more outcomes, i.e., a subset of the sample space. An event is considered to have "happened" during an experiment when the outcome of the latter is an element of the event. Since the same outcome may be a member of many events, it is possible for many events to have happened given a single outcome. For example, when the trial consists of throwing two dice, the set of all outcomes with a sum of 7 pips may constitute an event, whereas outcomes with an odd number of pips may constitute another event. If the outcome is the element of the elementary event of two pips on the first die and five on the second, then both of the events, "7 pips" and "odd number of pips", are said to have happened. * The probability measure $P$ is a function returning an event's probability. A probability is a real number between zero (impossible events have probability zero, though probability-zero events are not necessarily impossible) and one (the event happens almost surely, with total certainty). Thus $P$ is a function $P:\backslash mathcal\; \backslash rightarrow\; ()$. The probability measure function must satisfy two simple requirements: First, the probability of a countable union of mutually exclusive events must be equal to the countable sum of the probabilities of each of these events. For example, the probability of the union of the mutually exclusive events $\backslash text$ and $\backslash text$ in the random experiment of one coin toss, $P(\backslash text\backslash cup\backslash text)$, is the sum of probability for $\backslash text$ and the probability for $\backslash text$, $P(\backslash text)\; +\; P(\backslash text)$. Second, the probability of the sample space $\backslash Omega$ must be equal to 1 (which accounts for the fact that, given an execution of the model, some outcome must occur). In the previous example the probability of the set of outcomes $P(\backslash \backslash \})$ must be equal to one, because it is entirely certain that the outcome will be either $\backslash text$ or $\backslash text$ (the model neglects any other possibility) in a single coin toss. Not every subset of the sample space $\backslash Omega$ must necessarily be considered an event: some of the subsets are simply not of interest, others cannot be "measured". This is not so obvious in a case like a coin toss. In a different example, one could consider javelin throw lengths, where the events typically are intervals like "between 60 and 65 meters" and unions of such intervals, but not sets like the "irrational numbers between 60 and 65 meters" 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「Probability space」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.