Beta

The beta distribution, introduced in 1895 by Karl Pearson a famous British mathematician, was originally called the Pearson type 1 distribution. The name was changed in the 1940s to the beta distribution. Thomas Bayes also applied the distribution in 1763 as a posterior distribution to the parameter of the Bernoulli distribution. The beta distribution has many shapes that range from exponential, reverse exponential, right triangular, left triangular, skew right, skew left, normal and bathtub. The only fault is that it is a bit difficult to apply to real applications. The beta has two main parameters, k1 and k2 that are both larger than zero; and two location parameters a and b that define the limits of the admissible range. When the parameters are both larger than one, the beta variable is skewed to the left or to the right. These are the most used shapes of the distribution, and for this reason, the chapter concerns mainly these shapes. The random variable is denoted here as w where (a ≤ w ≤ b). A related variable, called the standard beta, x, has a range of 0–1. The mathematical properties of the probability density are defined with the standard beta. This includes the beta function and the gamma function, which are not easy to calculate. An algorithm is listed and needs to be calculated via a computed. There is no closed form solution to the cumulative probability, and thereby, a quantitative method is shown in the chapter examples. A straightforward process is used to convert from w to x and from x to w. When sample data is available, the sample average and mode are needed to estimate the parameter values of k1 and k2. A regression fit is developed to estimate the average of x from the mode of x. When sample data is not available, best estimates of the limits (a, b) and the most-likely value of w are obtained to estimate the parameter values.