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Realization (probability)
In probability and statistics, a realization, observation, or observed value, of a random variable is the value that is actually observed (what actually happened). The random variable itself is the process dictating how the observation comes about. Statistical quantities computed from realizations without deploying a statistical model are often called "empirical", as in empirical distribution function or empirical probability. Conventionally, to avoid confusion, upper case letters denote random variables; the corresponding lower case letters denote their realizations. Formal definition In more formal probability theory, a random variable is a function ''X'' defined from a sample space Ω to a measurable space called the state space. If an element in Ω is mapped to an element in state space by ''X'', then that element in state space is a realization. Elements of the sample space can be thought of as all the different possibilities that ''could'' happen; while a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', vol. 1, 3rd ed., (1968), Wiley, . This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formaliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Random Variate
In probability and statistics, a random variate or simply variate is a particular outcome or ''realization'' of a random variable; the random variates which are other outcomes of the same random variable might have different values ( random numbers). A random deviate or simply deviate is the difference of a random variate with respect to the distribution central location (e.g., mean), often divided by the standard deviation of the distribution (i.e., as a standard score). Random variates are used when simulating processes driven by random influences (stochastic processes). In modern applications, such simulations would derive random variates corresponding to any given probability distribution from computer procedures designed to create random variates corresponding to a uniform distribution, where these procedures would actually provide values chosen from a uniform distribution of pseudorandom numbers. Procedures to generate random variates corresponding to a given distribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Outcome (probability)
In probability theory, an outcome is a possible result of an Experiment (probability theory), experiment or trial. Each possible outcome of a particular experiment is unique, and different outcomes are Mutually exclusive events, mutually exclusive (only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a sample space. For the experiment where we flip a coin twice, the four possible ''outcomes'' that make up our ''sample space'' are (H, T), (T, H), (T, T) and (H, H), where "H" represents a "heads", and "T" represents a "tails". Outcomes should not be confused with Event (probability theory), events, which are (or informally, "groups") of outcomes. For comparison, we could define an event to occur when "at least one 'heads'" is flipped in the experiment - that is, when the outcome contains at least one 'heads'. This event would contain all outcomes in the sample space except the element (T, T). Sets of outco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Errors And Residuals
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error of an observation is the deviation of the observed value from the true value of a quantity of interest (for example, a population mean). The residual is the difference between the observed value and the '' estimated'' value of the quantity of interest (for example, a sample mean). The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals. In econometrics, "errors" are also called disturbances. Introduction Suppose there is a series of observations from a univariate distribution and we want to estimate the mean of that distribution (the so-called location model). In this case, the errors ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Elementary Event
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. 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 corresponding to precisely one outcome. The following are examples of elementary events: * All sets \, where k \in \N if objects are being counted and the sample space is S = \ (the natural numbers). * \, \, \, \text \ if a coin is tossed twice. S = \ where H stands for heads and T for tails. * All sets \, where x is a real number. Here X is a random variable with a normal distribution and S = (-\infty, + \infty). This example shows that, because the probability of each elementary event is zero, the probabilities assigned to elementary events do not determine a continuous probability distribution.. Probability of an elementary event Elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Event (probability Theory)
In probability theory, an event is a subset of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event consisting of only a single outcome is called an or an ; that is, it is a singleton set. An event that has more than one possible outcome is called a compound event. An event S is said to if S contains the outcome x of the experiment (or trial) (that is, if x \in S). The probability (with respect to some probability measure) that an event S occurs is the probability that S contains the outcome x of an experiment (that is, it is the probability that x \in S). An event defines a complementary event, namely the complementary set (the event occurring), and together these define a Bernoulli trial: did the event occur or not? Typically, when the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. When quantified, A \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that A \subseteq B, # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''. The validity of this technique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Map (mathematics)
In mathematics, a map or mapping is a function (mathematics), function in its general sense. These terms may have originated as from the process of making a map, geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but ''transformation (function), transformation'' often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. Maps as functions In many branches of mathematics, the term ''map'' is used to mean a Function (mathematics), function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Sample Space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, 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. They can also be finite, countably infinite, or uncountably infinite. A subset of the sample space is an event, denoted by E. If the outcome of an experiment is included in E, then event E has occurred. For example, if the experiment is tossing a single coin, the sample space is the set \, where the outcome H means that the coin is heads and the outcome T means that the coin is tails. The possible events are E=\, E=\, E = \, and E = \. For tossing two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |