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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.
This leads to different choices of sample space. The σ-algebra is a collection of all the events we would like to consider. This collection may or may not include each of the elementary events. Here, an "event" is a set of zero or more outcomes; that is, a subset of the sample space. An event is considered to have "happened" during an ...
In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set ... in advance, and uses constant space.
A random experiment is described or modeled by a mathematical construct known as a probability space. A probability space is constructed and defined with a specific kind of experiment or trial in mind. A mathematical description of an experiment consists of three parts: A sample space, Ω (or S), which is the set of all possible outcomes.
The sample space, often represented in notation by , is the set of all possible outcomes of a random phenomenon being observed. The sample space may be any set: a set of real numbers, a set of descriptive labels, a set of vectors, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip could be Ω = {"heads ...
A random variable is a measurable function: from a sample space as a set of possible outcomes to a measurable space. The technical axiomatic definition requires the sample space Ω {\displaystyle \Omega } to belong to a probability triple ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )} (see the measure-theoretic ...
In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space. [1] Using set theory terminology, an elementary event is a singleton. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event ...
A stochastic process is defined as a collection of random variables defined on a common probability space (,,), where is a sample space, is a -algebra, and is a probability measure; and the random variables, indexed by some set , all take values in the same mathematical space , which must be measurable with respect to some -algebra .