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 difference 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.