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We consider an independent and identically distributed (i.i.d.) randomdynamical system of simple linear transformations on the unit interval$T_{\beta}(x)=\beta x$ (mod $1$), $x\in[0,1]$, $\beta>0$, which are theso-called beta-transformations. For such a random dynamical system, includingthe case that it is generated by uncountably many maps, we give an explicitformula for the density function of a unique stationary measure under theassumption that the random dynamics is expanding in mean. As an application, inthe case that the random dynamics is generated by finitely many maps and themaps are chosen according to a Bernoulli measure, we show that the densityfunction is analytic as a function of parameter in the Bernoulli measure andgive its derivative explicitly. Furthermore, for a non-i.i.d. random dynamicalsystem of beta-transformations, we also give an explicit formula for the randomdensities of a unique absolutely continuous invariant measure under a certainstrong expanding condition or under the assumption that the maps randomlychosen are close to the beta-transformation for a non-simple number in thesense of parameter $\beta$.

Absolutely continuous invariant measures for random dynamical systems of beta-transformations