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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.
In ecology, alpha diversity (α-diversity) is the mean species diversity in a site at a local scale. The term was introduced by R. H. Whittaker [ 1 ] [ 2 ] together with the terms beta diversity (β-diversity) and gamma diversity (γ-diversity).
Gamma diversity and alpha diversity can be calculated directly from species inventory data. [2] [13] The simplest of Whittaker's original definitions of beta diversity is β = γ/α Here gamma diversity is the total species diversity of a landscape and alpha diversity is the mean species diversity per site.
In Bayesian statistics, the gamma distribution is widely used as a conjugate prior. It is the conjugate prior for the precision (i.e. inverse of the variance) of a normal distribution. It is also the conjugate prior for the exponential distribution.
Harmonic mean for Beta distribution for 0 < α < 5 and 0 < β < 5 (Mean - HarmonicMean) for Beta distribution versus alpha and beta from 0 to 2 Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), smaller values alpha and beta in front Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), larger values alpha and beta in front
It is a multivariate generalization of the beta distribution, [1] hence its alternative name of multivariate beta distribution (MBD). [2] Dirichlet distributions are commonly used as prior distributions in Bayesian statistics , and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial ...
In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind [1]) is an absolutely continuous probability distribution. If [,] has a beta distribution, then the odds has a beta prime distribution.
The term was introduced by R. H. Whittaker [1] together with the terms alpha diversity (α-diversity) and beta diversity (β-diversity). Whittaker's idea was that the total species diversity in a landscape (γ) is determined by two different things, the mean species diversity in sites at a more local scale (α) and the differentiation among ...