We consider several special cases of iterations of random i.i.d. linear functions with beta distributed fixed points that generate nested interval schemes when iterated in a backward direction, and ergodic Markov chains in the forward direction. We prove that the fixed points are beta distributed by using a related random equation with a gamma distributed solution that also generates a corresponding ergodic Markov chain with a gamma distributed stationary distribution. Our approach allows us to find limiting distributions of the random processes we consider, and provide solutions to the two random equations, by solving only one of these equations. The paper extends many partial results available in the existing literature.