Sample Space

In probability theory, the sample space or universal sample space, often denoted S, Ω, or U (for "universe"), of an experiment or random trial is the set of all possible outcomes. For example, if the experiment is tossing a coin, the sample space is the set {head, tail}. For tossing two coins, the sample space is {(head,head), (head,tail), (tail,head), (tail,tail)}. For tossing a single six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. For some kinds of experiments, there may be two or more plausible sample spaces available. For example, when drawing a card from a standard deck of 52 playing cards, one possibility for the sample space could be the rank (Ace through King), while another could be the suit (clubs, diamonds, hearts, or spades). A complete description of outcomes, however, would specify both the denomination and the suit, and a sample space describing each individual card can be constructed as the Cartesian product of the two sample spaces noted above.

In an elementary approach to probability, any subset of the sample space is usually called an event. However, this gives rise to problems when the sample space is infinite, so that a more precise definition of event is necessary. Under this definition only measurable subsets of the sample space, constituting a σ-algebra over the sample space itself, are considered events. However, this has essentially only theoretical significance, since in general the σ-algebra can always be defined to include all subsets of interest in applications.

Other articles related to "space, sample space":

Field Of Sets - Fields of Sets With Additional Structure - Sigma Algebras and Measure Spaces
... and the corresponding field of sets is called a measurable space ... The complexes of a measurable space are called measurable sets ... A measure space is a triple where is a measurable space and is a measure defined on it ...
Event (probability Theory)
... probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned ... Typically, when the sample space is finite, any subset of the sample space is an event (i.e ... all elements of the power set of the sample space are defined as events) ...
Categorical Distribution
... A categorical distribution is a discrete probability distribution whose sample space is the set of k individually identified items ... In one formulation of the distribution, the sample space is taken to be a finite sequence of integers ... In this formulation, the sample space can be considered to be the set of 1-of-K encoded random vectors x of dimension k having the property that exactly one ...
Probability Theory - Treatment - Motivation
... The collection of all results is called the sample space of the experiment ... The power set of the sample space is formed by considering all different collections of possible results ... Thus, the subset {1,3,5} is an element of the power set of the sample space of die rolls ...

Famous quotes containing the words space and/or sample:

    In the tale proper—where there is no space for development of character or for great profusion and variety of incident—mere construction is, of course, far more imperatively demanded than in the novel.
    Edgar Allan Poe (1809–1849)

    The present war having so long cut off all communication with Great-Britain, we are not able to make a fair estimate of the state of science in that country. The spirit in which she wages war is the only sample before our eyes, and that does not seem the legitimate offspring either of science or of civilization.
    Thomas Jefferson (1743–1826)