Randomly weighted averages with beta random proportions

A weighted average of two independent continuous random variables X1 and X2 with random proportions obtained by Beta distribution is introduced. A formula between the Stieltjes transforms of the distribution functions of the weighted averages and X1, X2 is established. We show that, among some other distributions, the Cauchy distribution and the power semicircle distribution can be characterized in a particular way by means of this construction.