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A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, [5] are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols.
Let (Ω, Σ, P) be a probability space.Let X : I × Ω → S be a stochastic process, where the index set I and state space S are both topological spaces.Then the process X is called sample-continuous (or almost surely continuous, or simply continuous) if the map X(ω) : I → S is continuous as a function of topological spaces for P-almost all ω in Ω.
A subset of the sample space of a procedure or experiment (i.e. a possible outcome) to which a probability can be assigned. For example, on rolling a die, "getting a three" is an event (with a probability of 1 ⁄ 6 if the die is fair), as is "getting a five or a six" (with a probability of 1 ⁄ 3).
This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a ...
For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements: [1] [2] A sample space, , which is the set of all possible outcomes. An event space, which is a set of events, , an event being a set of outcomes in the sample space.
That is, the probability function f(x) lies between zero and one for every value of x in the sample space Ω, and the sum of f(x) over all values x in the sample space Ω is equal to 1. An event is defined as any subset E {\displaystyle E\,} of the sample space Ω {\displaystyle \Omega \,} .
Chemical space is a concept in cheminformatics referring to the property space spanned by all possible molecules and chemical compounds adhering to a given set of construction principles and boundary conditions. It contains millions of compounds which are readily accessible and available to researchers.
A sample space must be the sample itself since all the sample points constitute a space, the sample or the sample space. The examples in the article here are actually "measurability" or scale space for the relevant experiments. They are not the samples of the experiments.