This is a very basic doubt from Statistical Mechanics which I haven't been able to resolve yet.
Suppose we take into account only single-particle states for a system, and we wish to look at the various possible scenarios with the help of Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein probability distributions. I've read that $f(E)$ is the probability distribution function, which also represents the average number of particles in a given state. While deriving this for the Bose-Einstein (B-E) or Fermi-Dirac (F-D) distributions, we've seen that $f(E)$ saturates at $1$ for F-D, while it can tend to $\infty$ for a B-E distribution. But is this true for a single-particle state as well(not single state!)? Should $f(E)$ lie in $[0,1]$ alike for distinguishable particles, Bosons and Fermions? Should $f(E)$ still represent an average number of particles for a single-particle state, as it's now a fraction? Can someone provide a detailed explanation to help me kill this doubt?