Number of bonds in the site-diluted lattices: sampling and fluctuations

We have calculated analytically the mean value and the variance of the number of bonds on the lattices of dimension $d$ for the given occupation of sites. We consider both kinds of site occupation: with the fixed concentration $n_s$ of occupied sites and with the probability $p$ for a site to be occupied. We found that the variance are smaller in the former case and not depends from the dimensionality of the hypercubic lattice. Whereas in the last case it grows with the space dimensionality. The ratio of variances are quite different in the limit of $p\to 1$. Finally, we demonstrate the relevance of the level of fluctuations on the precision of energy calculations for the Ising model in the Monte Carlo simulations.