The results for the negatively charged or all charged hadrons are similar to Eq. (\[positive\]) with summation over negative charges, $q_j<0,~q_k<0$, or all non-zero charges, $q_j\neq
0,~q_k\neq 0$, respectively. For $j\neq k$, one finds, $$\begin{aligned}
\label{jneqk}
\langle N_j  N_k \rangle ~=~ \frac{1}{Z_{id}} \left[\left(
\lambda_j
  \frac{\partial}{\partial \lambda_j} \right)~ \left( \lambda_k
  \frac{\partial}{\partial \lambda_k} \right)\right]~ Z_{id} ~=~ \langle
N_j \rangle ~\langle N_k \rangle~,\end{aligned}$$ and from Eqs. (\[DeltaNj\]) and (\[jneqk\]) it then follows, $$\begin{aligned}
\label{jkcorr}
\langle \Delta N_j \Delta N_k \rangle ~=~ \langle N_j  N_k \rangle
- \langle N_j \rangle ~\langle N_k \rangle ~=~ \delta_{jk}~
\langle N_j \rangle~.\end{aligned}$$ From Eq. (\[jkcorr\]) one concludes that in the ideal HG the correlations between different particle species are absent in the GCE. Any particle number distribution in the ideal Boltzmann gas is Poissonian (for large average multiplicity this is equivalent to a Gaussian) and its scaled variance therefore equals unity. For example, the scaled variance for positively charged hadrons is, $$\begin{aligned}
\label{omegaeq1}
\omega_+ ~=~ \frac{\langle \left( \Delta N_+ \right)^2
\rangle}{\langle N_+ \rangle}  ~=~ 1~.\end{aligned}$$ This is also valid for negatively charged and all charged hadrons.

Final State Fluctuations
------------------------

Final state yields and (co)variance have simple forms in the GCE [@Koch]: $$\begin{aligned}
\label{av-id}
\langle N_j \rangle ~&=~ 