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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.
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 ratio of the velocity of an object to the speed of light as used in the Lorentz factor
Lorentz factor γ as a function of fraction of given velocity and speed of light. Its initial value is 1 (when v = 0); and as velocity approaches the speed of light (v → c) γ increases without bound (γ → ∞). α (Lorentz factor inverse) as a function of velocity—a circular arc
In econometrics, the (α, θ) parameterization is common for modeling waiting times, such as the time until death, where it often takes the form of an Erlang distribution for integer α values. Bayesian statisticians prefer the ( α , λ ) parameterization, utilizing the gamma distribution as a conjugate prior for several inverse scale ...
The α-factor is a dimensionless quantity used to predict the solid–liquid interface type of a material during solidification. It was introduced by physicist Kenneth A. Jackson in 1958. In his model, crystal growth with larger values of α is smooth, whereas crystals growing at smaller α (below the threshold value of 2) have rough surfaces ...
Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b)) This result follows from the properties listed above. The incomplete beta function cannot be directly computed using such relations and other methods must be used. In GNU Octave, it is computed using a continued fraction expansion.
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.
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]