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In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
The positive integer values of the beta function are also the partial ... Flannery, BP (2007), "Section 6.1 Gamma Function, Beta Function, Factorials", Numerical ...
The Beta distribution on [0,1], a family of two-parameter distributions with one mode, of which the uniform distribution is a special case, and which is useful in estimating success probabilities. The four-parameter Beta distribution, a straight-forward generalization of the Beta distribution to arbitrary bounded intervals [,].
the beta coefficient, the non-diversifiable risk, of an asset in mathematical finance; the sideslip angle of an airplane; a beta particle (e − or e +) the beta brain wave in brain or cognitive sciences; ecliptic latitude in astronomy; the ratio of plasma pressure to magnetic pressure in plasma physics; β-reduction in lambda calculus
The beta of an asset is compared to the market as a whole, usually the S&P 500. By definition, the value-weighted average of all market-betas of all investable assets with respect to the value-weighted market index is 1. If an asset has a beta above 1, it indicates that its return moves more than 1-to-1 with the return of the market-portfolio ...
The base e is the most economical choice of radix β > 1, [4] where the radix economy is measured as the product of the radix and the length of the string of symbols needed to express a given range of values.
The mass of probability distribution is balanced at the expected value, here a Beta(α,β) distribution with expected value α/(α+β). In classical mechanics, the center of mass is an analogous concept to expectation. For example, suppose X is a discrete random variable with values x i and corresponding probabilities p i.
The beta-binomial distribution can also be motivated via an urn model for positive integer values of α and β, known as the Pólya urn model. Specifically, imagine an urn containing α red balls and β black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn.