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The regularized incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function (;,) of a random variable X following a binomial distribution with probability of single success p and number of Bernoulli trials n:
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.
For every odd positive integer +, the following equation holds: [3] (+) = ()!() +where is the n-th Euler Number.This yields: =,() =,() =,() =For the values of the Dirichlet beta function at even positive integers no elementary closed form is known, and no method has yet been found for determining the arithmetic nature of even beta values (similarly to the Riemann zeta function at odd integers ...
where β is the Dirichlet beta function. Its numerical value [1] is approximately (sequence A006752 in the OEIS) G = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 … Catalan's constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865. [2] [3]
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.
The beta negative binomial distribution contains the beta geometric distribution as a special case when either = or =. It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large α {\displaystyle \alpha } .
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 ]
where I is the regularized incomplete beta function. While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds.