How can the total number of particles can be sum over all Density of states?

Question

It is known that Density of states of a lattice structure is the probability distribution function of energy (for reference). This means that the sum/Integral of DOS over all energies is $1$. However, we also know the relation for $T \rightarrow 0$, $$N=\int_{-\infty}^{E_F}D(E)dE$$ where, N is number of particles/atoms in a unit cell. My question is how can $N>1$ when the total sum is $1$. (Note that N can be more that zero for example there are $2$ atoms in the unit cell of BCC)

Answer 1

So, I asked the same question to my research advisor and he said that in general in the field of condensed matter, we don't consider DOS as PDF. It is just number of states between $E_0$ and $E_0+\Delta E$. The total integral is not necessarily normalized. One can think it as non-normalized PDF.