Quantum particles are characterized, for a given preparation, by a fundamental stochasticity of their observable features, such as positions, momenta, energies, or times, e.g., times of arrival at a detector in time-of-flight experiments. In quantum theory the preparation stage is encoded in a wave function, whereas averages or statistical moments of the observables are calculated by a well-known prescription (the expectation value integral) from self-adjoint operators and their powers: this is at least the case for position, momentum, or energy. In fact, the entire statistical distributions are given by the square modulus of the overlap of the wave function with the corresponding eigenstates.