  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. 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 this new
transform and also integral representation of the Gauss-hypergeometric function
in some parts are employed to achieve the exact distribution. Several examples
of the new family are investigated.
