  Some exact formulae of the expectation values and probability densities in a
weak measurement for an operator ${\bf A}$ which satisfies the property ${\bf
A}^{2}=1$ are derived. These formulae include all-order effects of the unitary
evolution due to the von-Neumann interaction. These are valid not only in the
weak measurement regime but also in the strong measurement regime and tell us
the connection between these two regime. Using these formulae, arguments of the
optimization of the signal amplification and the signal to noise ratio are
developed in two typical experimental setups.
