  The exact statistics of an arbitrary quantum observable is analytically
obtained. Due to the probabilistic nature of a sequence of intermediate
measurements and stochastic fluctuations induced by the interaction with the
environment, the measurement outcomes at the end of the system's evolution are
random variables. Here, we provide the exact large-deviation form of their
probability distribution, which is given by an exponentially decaying profile
in the number of measurements. The most probable distribution of the
measurement outcomes in a single realization of the system transformation is
then derived, thus achieving predictions beyond the expectation value. The
theoretical results are confirmed by numerical simulations of an experimentally
reproducible two-level system with stochastic Hamiltonian.
