  Within the path integral formalism, we derive exact expressions for
correlation functions measuring the lattice charge induced by an electron and
associated polarization in Frohlich polaron problem. We prove that a sum rule
for the total induced charge, already obtained within approximated approaches
is indeed exact. As a consequence the total induced charge is shown rigorously
to be temperature independent. In addition we perform path integral Monte Carlo
calculations of the correlation functions and we compare with variational
results based on
  Feynman method. As the temperature increases the polaron radius decreases. On
the other hand at high temperatures the electron motion is not hindered by the
lattice. These apparently contradictory results are discussed.
