Invariant density functions of random β-transformations

We consider the random -transformation introduced by Dajani and Kraaikamp [Random -expansions. Ergod. Th. & Dynam. Sys. 23 (2003), 461-479], which is defined on . We give an explicit formula for the density function of a unique -invariant probability measure absolutely continuous with respect to the product measure , where is the -Bernoulli measure on and is the normalized Lebesgue measure on . We apply the explicit formula for the density function to evaluate its upper and lower bounds and to investigate its continuity as a function of the two parameters and .