  We consider a type of intermittent behavior that occurs as the result of the
interplay between dynamical mechanisms giving rise to type-I intermittency and
random dynamics. We analytically deduce the laws for the distribution of the
laminar phases, with the law for the mean length of the laminar phases versus
the critical parameter deduced earlier [PRE 62 (2000) 6304] being the corollary
fact of the developed theory. We find a very good agreement between the
theoretical predictions and the data obtained by means of both the experimental
study and numerical calculations. We discuss also how this mechanism is
expected to take place in other relevant physical circumstances.
