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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 has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines.
The beta distribution is a suitable model for the random behavior of percentages and proportions.
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In Bayesian inference , the beta distribution is the conjugate prior probability distribution for the Bernoulli , binomial , negative binomial , and geometric distributions. The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind , whereas beta distribution of the second kind is an alternative name for the beta prime distribution.
The generalization to multiple variables is called a Dirichlet distribution. Several authors, including N. Johnson and S. The cumulative distribution function is. For sample size much larger than 2, the difference between these two priors becomes negligible.
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See section Bayesian inference for further details. This parametrization may be useful in Bayesian parameter estimation.
For example, one may administer a test to a number of individuals. The mean and sample size parameters are related to the shape parameters α and β via [ 3 ]. Solving the system of coupled equations given in the above sections as the equations for the mean and the variance of the beta distribution in terms of the original parameters α and β , one can express the α and β parameters in terms of the mean μ and the variance var :.
This parametrization of the beta distribution may lead to a more intuitive understanding than the one based on the original parameters α and β.
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For example, by expressing the mode, skewness, excess kurtosis and differential entropy in terms of the mean and the variance:. A beta distribution with the two shape parameters α and β is supported on the range [0,1] or 0,1. That a random variable Y is beta-distributed with four parameters α, β, a , and c will be denoted by:. Note: the geometric mean and harmonic mean cannot be transformed by a linear transformation in the way that the mean, median and mode can.
Since the skewness and excess kurtosis are non-dimensional quantities as moments centered on the mean and normalized by the standard deviation , they are independent of the parameters a and c , and therefore equal to the expressions given above in terms of X with support [0,1] or 0,1 :. See Shapes section in this article for a full list of mode cases, for arbitrary values of α and β.
For several of these cases, the maximum value of the density function occurs at one or both ends. In some cases the maximum value of the density function occurring at the end is finite.
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In several other cases there is a singularity at one end, where the value of the density function approaches infinity. There is no general closed-form expression for the median of the beta distribution for arbitrary values of α and β. Closed-form expressions for particular values of the parameters α and β follow: [ citation needed ]. The following are the limits with one parameter finite non-zero and the other approaching these limits: [ citation needed ].
A reasonable approximation of the value of the median of the beta distribution, for both α and β greater or equal to one, is given by the formula [ 9 ]. The absolute error divided by the difference between the mean and the mode is similarly small:. Also, the following limits can be obtained from the above expression:. Following are the limits with one parameter finite non-zero and the other approaching these limits:.
As Mosteller and Tukey remark [ 10 ] p. This illustrates how, for short-tailed distributions, the extreme observations should get more weight. The logarithm of the geometric mean G X of a distribution with random variable X is the arithmetic mean of ln X , or, equivalently, its expected value:. Therefore, the geometric mean of a beta distribution with shape parameters α and β is the exponential of the digamma functions of α and β as follows:.
The accompanying plot shows the difference between the mean and the geometric mean for shape parameters α and β from zero to 2.