Generalized beta distribution

In probability and statistics, the generalized beta distribution is a continuous probability distribution with five parameters, including more than thirty named distributions as limiting or special cases. It has been used in the modeling of income distribution, stock returns, as well as in regression analysis. The exponential generalized Beta (EGB) distribution follows directly from the GB and generalizes other common distributions. In probability and statistics, the generalized beta distribution is a continuous probability distribution with five parameters, including more than thirty named distributions as limiting or special cases. It has been used in the modeling of income distribution, stock returns, as well as in regression analysis. The exponential generalized Beta (EGB) distribution follows directly from the GB and generalizes other common distributions. A generalized beta random variable, Y, is defined by the following probability density function: and zero otherwise. Here the parameters satisfy 0 ≤ c ≤ 1 {displaystyle 0leq cleq 1} and b {displaystyle b} , p {displaystyle p} , and q {displaystyle q} positive. The function B(p,q) is the beta function. It can be shown that the hth moment can be expressed as follows: where 2 F 1 {displaystyle {}_{2}F_{1}} denotes the hypergeometric series (which converges for all h if c<1, or for all h/a<q if c=1 ).