Expectation Values in the Quantum Trajectory Formalism

If I want to know the expectation value of an operator O in the quantum trajectory formalism, I average over $N$ trajectories, where I call one such trajectory $Psi_n$:

begin{equation}
langle O rangle = frac{1}{N}sum_{n=1}^N langle Psi_n | O |Psi_n rangle,
end{equation}
correct?

If so, my question is: Is that still correct if the different $Psi_n$ are not orthogonal/parallel to each other?

For example: If I am interested in the average probabiltiy density of the position distribution $rho$, do I get it by calculating:

begin{equation}
rho(x) = frac{1}{N}sum_{n=1}^N Psi^*_n(x) Psi_n(x),
end{equation}

even if $langle Psi_n(x)|Psi_m(x) rangle neq delta_{n,m} $?

If not, what would be the correct formula?