Beta distribution

any value in ( 0 , 1 ) {displaystyle (0,1)} for α, β = 1In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval parametrized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. It is a special case of the Dirichlet distribution. In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval parametrized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. It is a special case of the Dirichlet 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. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. For example, the beta distribution can be used in Bayesian analysis to describe initial knowledge concerning probability of success such as the probability that a space vehicle will successfully complete a specified mission. The beta distribution is a suitable model for the random behavior of percentages and proportions. The usual formulation of the beta distribution 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 probability density function (pdf) of the beta distribution, for 0 ≤ x ≤ 1, and shape parameters α, β > 0, is a power function of the variable x and of its reflection (1 − x) as follows: where Γ(z) is the gamma function. The beta function, B {displaystyle mathrm {B} } , is a normalization constant to ensure that the total probability is 1. In the above equations x is a realization—an observed value that actually occurred—of a random process X. This definition includes both ends x = 0 and x = 1, which is consistent with definitions for other continuous distributions supported on a bounded interval which are special cases of the beta distribution, for example the arcsine distribution, and consistent with several authors, like N. L. Johnson and S. Kotz. However, the inclusion of x = 0 and x = 1 does not work for α, β < 1; accordingly, several other authors, including W. Feller, choose to exclude the ends x = 0 and x = 1, (so that the two ends are not actually part of the domain of the density function) and consider instead 0 < x < 1. Several authors, including N. L. Johnson and S. Kotz, use the symbols p and q (instead of α and β) for the shape parameters of the beta distribution, reminiscent of the symbols traditionally used for the parameters of the Bernoulli distribution, because the beta distribution approaches the Bernoulli distribution in the limit when both shape parameters α and β approach the value of zero. In the following, a random variable X beta-distributed with parameters α and β will be denoted by: