  The tomographic invertable map of the Wigner function onto the positive
probability distribution function is studied. Alternatives to the Schr\"odinger
evolution equation and to the energy level equation written for the positive
probability distribution are discussed. Instead of the transition probability
amplitude (Feynman path integral) a transition probability is introduced.
  A new formulation of the conventional quantum mechanics (without wave
function and density matrix) based on the ``probability representation'' of
quantum states is given. An equation for the propagator in the new formulation
of quantum mechanics is derived. Some paradoxes of quantum mechanics are
reconsidered.
