Elliptical Distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms an ellipse and an ellipsoid, respectively, in iso-density plots. In statistics, the normal distribution is used in ''classical'' multivariate analysis, while elliptical distributions are used in ''generalized'' multivariate analysis, for the study of symmetric distributions with tails that are heavy, like the multivariate t-distribution, or light (in comparison with the normal distribution). Some statistical methods that were originally motivated by the study of the normal distribution have good performance for general elliptical distributions (with finite variance), particularly for spherical distributions (which are defined below). Elliptical distributions are also used in robust statistics to evaluate proposed multiv ...
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Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."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, . The higher the probability of an event, the more likely it is that the event will occur. 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 ...
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Pseudo-random Number
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Background The generation of random numbers has many uses, such as for random sampling, Monte Carlo methods, board games, or gambling. In physics, however, most processes, such as gravitational acceleration, are deterministic, meaning that they always produce the same outcome from the same starting point. Some notable exceptions are radioactive decay and quantum measurement, which are both modeled as being truly random processes in the underlying physics. Since these processes are not practical sources of random numbers, people use pseudorandom numbers, which ideally have the unpredictability of a truly random sequence, despite being generated by a deterministic process. In many applications, the deterministic process is a computer algorithm called a pseudorandom number generator, which must first be provided wit ...
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Multivariate Logistic Distribution
Multivariate may refer to: In mathematics * Multivariable calculus * Multivariate function * Multivariate polynomial In computing * Multivariate cryptography * Multivariate division algorithm * Multivariate interpolation * Multivariate optical computing * Multivariate optimization, used for the design of heat exchangers, see In statistics * Multivariate analysis * Multivariate random variable * Multivariate statistics See also * Univariate * Bivariate (other) Bivariate may refer to: Mathematics * Bivariate function, a function of two variables * Bivariate polynomial, a polynomial of two indeterminates Statistics * Bivariate data, that shows the relationship between two variables * Bivariate analys ...

Multivariate Laplace Distribution
In the mathematical theory of probability, multivariate Laplace distributions are extensions of the Laplace distribution and the asymmetric Laplace distribution to multiple variables. The marginal distributions of symmetric multivariate Laplace distribution variables are Laplace distributions. The marginal distributions of asymmetric multivariate Laplace distribution variables are asymmetric Laplace distributions. Symmetric multivariate Laplace distribution A typical characterization of the symmetric multivariate Laplace distribution has the characteristic function: : \varphi(t;\boldsymbol\mu,\boldsymbol\Sigma) = \frac, where \boldsymbol\mu is the vector of means for each variable and \boldsymbol\Sigma is the covariance matrix. Unlike the multivariate normal distribution, even if the covariance matrix has zero covariance and correlation the variables are not independent. The symmetric multivariate Laplace distribution is elliptical. Probability density function If \bolds ...
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Multivariate Stable Distribution
The multivariate stable distribution is a multivariate probability distribution that is a multivariate generalisation of the univariate stable distribution. The multivariate stable distribution defines linear relations between stable distribution marginals. In the same way as for the univariate case, the distribution is defined in terms of its characteristic function. The multivariate stable distribution can also be thought as an extension of the multivariate normal distribution. It has parameter, ''α'', which is defined over the range 0 < ''α'' ≤ 2, and where the case ''α'' = 2 is equivalent to the multivariate normal distribution. It has an additional skew parameter that allows for non-symmetric distributions, where the is sym ...
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Multivariate T-distribution
In statistics, the multivariate ''t''-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's ''t''-distribution, which is a distri