Average number of particles in a certain energy level in the Canonical Ensemble

A quantum system has $r$ discrete energy levels $varepsilon_1,varepsilon_2,varepsilon_3,…,varepsilon_r$ and $N$ particles distributed in these levels, with the number of particles at each level denoted by $n_1,n_2,n_3,…,n_r$. I’m trying to find the average number of particles in the $i$-th energy level, $leftlangle n_irightrangle$, and the fluctuation of this average, $leftlangle(Delta n_i)^{2}rightrangle$, using the Canonical Ensemble.

My attempt

The average energy of the system at the state $R$ determined by the occupation numbers $(n_1,n_2,n_3,…,n_r)_R$ can be computed by

$$
langle Erangle=leftlangle E_{R}rightrangle=sum_{R} P_{R} E_{R}
=frac{1}{Z}sum_{R} E_{R} e^{-beta E_{R}}
=-frac{1}{Z}bigg(frac{partial Z}{partial beta}bigg)_{N, V}
=-bigg(frac{partial ln Z }{partial beta}bigg)_{N, V}
$$

With a similar process, keeping in mind that $E_{R} = sum_{r} n_r varepsilon_{r}$, one gets that

$$langle n_irangle = sum_{R} P_{R} n_i
=frac{1}{Z}sum_{R} n_r e^{-beta sum_{r} n_i varepsilon_{r}}
=-frac{1}{beta}bigg(frac{partial ln Z}{partial varepsilon_i}bigg)_{N, V}
$$

Which is supposed to be the correct result. However, I am not sure that this $langle n_i rangle = sum_{R} P_{R} n_i$ is valid for this average since $P_r$ is the probability that the system is in the $R$-state, not that the $r$-th energy level has a certain number of particles…

Is the procedure I have performed in this correct?