  We examine weak measurements of arbitrary observables where the object is
prepared in a mixed state and on which measurements with imperfect detectors
are made. The weak value of an observable can be expressed as a conditional
expectation value over an infinite class of different generalized Kirkwood
quasi-probability distributions. "Strange" weak values for which the real part
exceeds the eigenvalue spectrum of the observable can only be found if the
Terletsky-Margenau-Hill distribution is negative, or, equivalently, if the real
part of the weak value of the density operator is negative. We find that a
classical model of a weak measurement exists whenever the
Terletsky-Margenau-Hill representation of the observable equals the classical
representation of the observable and the Terletsky-Margenau-Hill distribution
is nonnegative. Strange weak values alone are not sufficient to obtain a
contradiction with classical models.
  We propose feasible weak measurements of photon number of the radiation
field. Negative weak values of energy contradicts all classical stochastic
models, whereas negative weak values of photon number contradict all classical
stochastic models where the energy is bounded from below by the zero-point
energy. We examine coherent states in particular, and find negative weak values
with probabilities of 16% for kinetic energy (or squared field quadrature), 8%
for harmonic oscillator energy and 50% for photon number. These experiments are
robust against detector inefficiency and thermal noise.
