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We give the exact distribution of the average of n independent beta randomvariables weighted by the selected cuts of (0, 1) by the order statistics of arandom sample of size n-1 from the uniform distribution U(0,1), for each n. Anew integral transformation that is similar to generalized Stieltjes transformis given with various properties. The result of Soltani and Roozegar [Ondistribution of randomly ordered uniform incremental weighted averages: Divideddifference approach. Statist Probab Lett. 2012, 82(5):1012-1020] with this newtransform and also integral representation of the Gauss-hypergeometric functionin some parts are employed to achieve the exact distribution. Several examplesof the new family are investigated.

Certain Family of Some Beta Distributions Arising from Distribution of Randomly Weighted Average