If A=A(X), i.e., a function of the operator X only, then the trace can be easily evaluated in the coordinate basis:
Since A(X) acts to the left on one of its eigenstates, we have
which only involves a diagonal element of the density matrix. This can, therefore, be written as a path integral:
However, since all points 
are equivalent, due to the fact that
they are all integrated over, we can make P equivalent cyclic
renaming of the coordinates 
, 
, etc.
and generate P equivalent integrals. In each, the function 
or 
, etc. will appear. If we sum these P equivalent integrals
and divide by P, we get an expression:
This allows us to define an estimator for the observable A.
Recall that an estimator is a function of the P variables 
whose average over the ensemble yields the expectation value of A:
Then
where the average on the right is taken over many configurations
of the P variables (we will discuss, in the nex lecture,
a way to generate these configurations).
The limit 
can be taken in the same way that
we did in the previous lecture, yielding a functional integral
expression for the expectation value:
