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Constant Random Variable
In probability theory, a degenerate distribution on a measure space (E, \mathcal, \mu) is a probability distribution whose support is a null set with respect to \mu. For instance, in the -dimensional space endowed with the Lebesgue measure, any distribution concentrated on a -dimensional subspace with is a degenerate distribution on . This is essentially the same notion as a singular probability measure, but the term ''degenerate'' is typically used when the distribution arises as a limit of (non-degenerate) distributions. When the support of a degenerate distribution consists of a single point , this distribution is a Dirac measure in : it is the distribution of a deterministic random variable equal to with probability 1. This is a special case of a discrete distribution; its probability mass function equals 1 in and 0 everywhere else. In the case of a real-valued random variable, the cumulative distribution function of the degenerate distribution localized in is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
WikiProject Probability
A WikiProject, or Wikiproject, is an affinity group for contributors with shared goals within the Wikimedia movement. WikiProjects are prevalent within the largest wiki, Wikipedia, and exist to varying degrees within sibling projects such as Wiktionary, Wikiquote, Wikidata, and Wikisource. They also exist in different languages, and translation of articles is a form of their collaboration. During the COVID-19 pandemic, CBS News noted the role of Wikipedia's WikiProject Medicine in maintaining the accuracy of articles related to the disease. Another WikiProject that has drawn attention is WikiProject Women Scientists, which was profiled by '' Smithsonian'' for its efforts to improve coverage of women scientists which the profile noted had "helped increase the number of female scientists on Wikipedia from around 1,600 to over 5,000". On Wikipedia Some Wikipedia WikiProjects are substantial enough to engage in cooperative activities with outside organizations relevant to the fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Distributions
Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a group with the discrete topology *Discrete category, category whose only arrows are identity arrows *Discrete mathematics, the study of structures without continuity *Discrete optimization, a branch of optimization in applied mathematics and computer science *Discrete probability distribution, a random variable that can be counted *Discrete space, a simple example of a topological space *Discrete spline interpolation, the discrete analog of ordinary spline interpolation *Discrete time, non-continuous time, which results in discrete-time samples *Discrete variable In mathematics and statistics, a quantitative variable may be continuous or discrete. If it can take on two real values and all the values between them, the variable is con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Distribution
In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function (for continuous probability distribution) or probability mass function (for discrete random variables) is reflected around a vertical line at some value of the random variable represented by the distribution. This vertical line is the line of symmetry of the distribution. Thus the probability of being any given distance on one side of the value about which symmetry occurs is the same as the probability of being the same distance on the other side of that value. Formal definition A probability distribution is said to be symmetric if and only if there exists a value x_0 such that : f(x_0-\delta) = f(x_0+\delta) for all real numbers \delta , where ''f'' is the probability density function if the distribution is continuous or the probability mass function if the distribution is discre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Probability Distribution
A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw-Hill Connect. Webp.274/ref> They are constructed to allow for different degrees and types of movement. Some joints, such as the knee, elbow, and shoulder, are self-lubricating, almost frictionless, and are able to withstand compression and maintain heavy loads while still executing smooth and precise movements. Other joints such as suture (joint), sutures between the bones of the skull permit very little movement (only during birth) in order to protect the brain and the sense organs. The connection between a tooth and the jawbone is also called a joint, and is described as a fibrous joint known as a gomphosis. Joints are classified both structurally and functionally. Joints play a vital role in the human body, contributing to movement, sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Semidefinite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \mathbf^* M \mathbf is positive for every nonzero complex column vector \mathbf, where \mathbf^* denotes the conjugate transpose of \mathbf. Positive semi-definite matrices are defined similarly, except that the scalars \mathbf^\mathsf M \mathbf and \mathbf^* M \mathbf are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''. Some authors use more general definitions of definiteness, permitting the matrices to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance Matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2 \times 2 matrix would be necessary to fully characterize the two-dimensional variation. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). The covariance matrix of a random vector \mathbf is typically denoted by \operatorname_, \Sigma or S. Definition Throughout this article, boldfaced u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Distribution
Multivariate is the quality of having multiple variables. It may also refer to: In mathematics * Multivariable calculus * Multivariate function * Multivariate polynomial * Multivariate interpolation * Multivariate optimization In computing * Multivariate cryptography * Multivariate division algorithm * Multivariate optical computing In statistics * Multivariate analysis * Multivariate random variable * Multivariate regression * Multivariate statistics See also * Univariate In mathematics, a univariate object is an expression (mathematics), expression, equation, function (mathematics), function or polynomial involving only one Variable (mathematics), variable. Objects involving more than one variable are ''wikt:multi ... * Bivariate (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Constant Function
In mathematics, a constant function is a function whose (output) value is the same for every input value. Basic properties As a real-valued function of a real-valued argument, a constant function has the general form or just For example, the function is the specific constant function where the output value is . The domain of this function is the set of all real numbers. The image of this function is the singleton set . The independent variable does not appear on the right side of the function expression and so its value is "vacuously substituted"; namely , , , and so on. No matter what value of is input, the output is . The graph of the constant function is a ''horizontal line'' in the plane that passes through the point . In the context of a polynomial in one variable , the constant function is called ''non-zero constant function'' because it is a polynomial of degree 0, and its general form is , where is nonzero. This function has no intersection point with the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
