  This paper presents a one shot analysis of the lossy compression problem
under average distortion constraints. We calculate the exact expected
distortion of a random code. The result is given as an integral formula using a
newly defined functional $\tilde{D}(z,Q_Y)$ where $Q_Y$ is the random coding
distribution and $z\in [0,1]$. When we plug in the code distribution as $Q_Y$,
this functional produces the average distortion of the code, thus provide a
converse result utilizing the same functional. Two alternative formulas are
provided for $\tilde{D}(z,Q_Y)$, the first involves a supremum over some
auxiliary distribution $Q_X$ which has resemblance to the channel coding
meta-converse and the other involves an infimum over channels which resemble
the well known Shannon distortion-rate function.
