Concepts inInverting the symmetrical beta distribution
Beta distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval (0, 1) parameterized by two positive shape parameters, typically denoted by ¿ and ¿. The beta distribution can be suited to the statistical modelling of proportions in applications where values of proportions equal to 0 or 1 do not occur.
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Symmetry
Symmetry (from Greek ¿¿¿¿¿¿¿¿¿¿ symmetría "measure together") generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise.
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Gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. There are two different parameterizations in common use: With a shape parameter k and a scale parameter ¿. With a shape parameter ¿ = k and an inverse scale parameter ¿ = ¿¿, called a rate parameter. The parameterization with k and ¿ appears to be more common in econometrics and certain other applied fields, where e.g.
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Random variate
A random variate is a particular outcome of a random variable: the random variates which are other outcomes of the same random variable would have different values. Random variates are used when simulating processes driven by random influences.
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Invertible matrix
In linear algebra an n-by-n (square) matrix A is called invertible (some authors use nonsingular or nondegenerate) if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A, denoted by A.
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Newton's method
In numerical analysis, Newton's method (also known as the Newton¿Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.
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Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values. For a more precise definition one needs to distinguish between discrete and continuous random variables. In the discrete case, one can easily assign a probability to each possible value: when throwing a die, each of the six values 1 to 6 has the probability 1/6.
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