Limiting behavior of the probability density

In the grand canonical ensemble, the probability of finding N molecules in a (constant) volume V at some fixed temperature T may be cast as 

However, the fact that the first-order term in the Taylor expansion vanishes equips us with an additional relation, namely 

With the aid of P N) we can also calculate the average number of molecules via 

To analyze the simulation data in Section (5.4.2) for the isothermal compressibility, it turns out to be more convenient to transform variables in this last expression according to N, (N) p,p to obtain an analogous expression for the density distribution P (p) in a. slit-pore. 

An alternative route to Elq. (C.36) proceeds as follows. Suppose wo take a system of constant volume at fixed T and p. Let us perform an experiment on this s tem that allows us to determine the particle number N. As under conditions of fixed T and p, but where N may vary, there is a certain probability p that the result of our measxu-ement will be positive, that is, it will give precisely N. Clearly, the outcome of the experiment may also be negative that is, there is a probability - p that we will measure some 

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