Random oracles and independence
If $H(X)$ and $H(Y)$ were not evaluated as a part of selecting $X$ and $Y$, then yes; the assumption of a random Oracle is that $H(Z)$ is an independent and uniformly distributed variable for every new (not previously submitted to the Oracle) value $Z$.

If this isn't the case, then this need not hold.  One obvious counterexample is:

$X := 1; Y := 2 \textit{ if H(1) &lt; H(2)}\\ X := 2; Y :=1 \textit{ otherwise}$