  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.
