Bit Error Probability for 8PSK
Equations (1) and (2) don't appear to be correct, or at least they are incomplete.  The received signal will be a 2 dimensional gaussian centered on the (000) signal point or (x = 1, y = 0).  That is:



So to compute the error probability, you integrate f(x,y) over the regions you describe above.

The noise vector, n, is a gaussian random vector and is not generally normal to the decision boundaries.