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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 function is also important in statistics, e.g. for the beta distribution and beta prime distribution. As briefly alluded to previously, the beta function is closely tied with the gamma function and plays an important role in calculus.
The distribution has important applications in various fields, including econometrics, Bayesian statistics, life testing. [3] In econometrics, the ( α , θ ) parameterization is common for modeling waiting times, such as the time until death, where it often takes the form of an Erlang distribution for integer α values.
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
For a Type I error, it is shown as α (alpha) and is known as the size of the test and is 1 minus the specificity of the test. This quantity is sometimes referred to as the confidence of the test, or the level of significance (LOS) of the test. For a Type II error, it is shown as β (beta) and is 1 minus the power or 1 minus the sensitivity of ...
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, [2] is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend ...
Wire-grid Cobb–Douglas production surface with isoquants A two-input Cobb–Douglas production function with isoquants. In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly physical capital and labor) and ...
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 c = 0 {\displaystyle c=0} , b = 1 {\displaystyle b=1} yields the standard two-parameter beta distribution .