In quantum mechanics, the probability distribution function and full counting statistics play a fundamental role in characterizing the fluctuations of quantum observables, as they encode the complete information about these fluctuations. In this Letter, we measure these two quantities in a trapped-ion quantum simulator for the transverse and longitudinal magnetization within a subsystem. We utilize the toolbox of classical shadows to postprocess the measurements performed in random bases. The measurement scheme efficiently allows access to the full counting statistics and probability distribution function of all possible operators on desired choices of subsystems of an extended quantum system.
