Exact Distribution of Random Weighted Convolution of Some Beta Distributions Through an Integral Transform

We give the exact distribution of the average of n independent beta random variables weighted by the selected cuts of (0; 1) by the order statistics of a random sample of size n−1 from the uniform distribution U(0; 1), for each n. A new integral transformation that is similar to generalized Stieltjes transform is given with various properties. Integral representation of the Gauss-hypergeometric function in some parts is employed to achieve the exact distribution. Also the result of Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: Divided dierence approach. Statist Probab Lett. 2012;82(5):1012{1020] with the new transform is achieved. Finally, several new examples of this family of distributions are investigated