  We study for the first time linear response for random compositions of maps,
chosen independently according to a distribution $\PP$. We are interested in
the following question: how does an absolutely continuous stationary measure
(acsm) of a random system change when $\PP$ changes smoothly to $\PP_{\eps}$?
For a wide class of one dimensional random maps, we prove differentiability of
acsm with respect to $\eps$; moreover, we obtain a linear response formula. We
apply our results to iid compositions, with respect to various distributions
$\PP_{\eps}$, of uniformly expanding circle maps, Gauss-R\'enyi maps (random
continued fractions) and Pomeau-Manneville maps. Our results yield an exact
formula for the invariant density of random continued fractions; while for
Pomeau-Manneville maps our results provide a precise relation between their
linear response under certain random perturbations and their linear response
under deterministic perturbations.
