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