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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 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 family includes the beta of the first and second kind [7] (B1 and B2, where the B2 is also referred to as the Beta prime), which correspond to c = 0 and c = 1, respectively. Setting =, = yields the standard two-parameter beta distribution.
|fisher= — the Fisher information matrix for the model. |KLDiv= — the Kullback-Leibler divergence of the model |JSDiv= — the Jensen-Shannon divergence of the model |moments= — formulas to use in Method of moments for the model. |ES= — the Expected shortfall or CVaR for the model. |bPOE= — the Buffered Probability of Exceedance for ...
Some distributions have been specially named as compounds: beta-binomial distribution, Beta negative binomial distribution, gamma-normal distribution. Examples: If X is a Binomial(n,p) random variable, and parameter p is a random variable with beta(α, β) distribution, then X is distributed as a Beta-Binomial(α,β,n).
The Type I cumulative distribution function is usually represented as a Poisson mixture of central beta random variables: [1] = = (+,),where λ is the noncentrality parameter, P(.) is the Poisson(λ/2) probability mass function, \alpha=m/2 and \beta=n/2 are shape parameters, and (,) is the incomplete beta function.
Beta regression is a form of regression which is used when the response variable, , takes values within (,) and can be assumed to follow a beta distribution. [1] It is generalisable to variables which takes values in the arbitrary open interval ( a , b ) {\displaystyle (a,b)} through transformations. [ 1 ]
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