I was reading Statistical Physics of Particles by Mehran Kardar. In unit 3.3 of chapter 3 named Kinetic theory of gases, the author derived the BBGKY hierarchy. Now in the derivation, the author defines one-particle density $f_1(\vec{p}, \vec{q}, t)$ as the expectation value of finding any of the $N$ particles at location $\vec{q}$, with momentum $\vec{p}$, at time $t$. Then the author calculates the one-particle density using phase space density $\rho(\vec{p}_1, \vec{q}_1, \vec{p}_2, \vec{q}_2, \cdots, \vec{p}_N, \vec{q}_N, t)$ as $$f_1(\vec{p}, \vec{q}, t)=\left\langle\sum_{i=1}^N\delta^3(\vec{p}-\vec{p_i})\delta^3(\vec{q}-\vec{q_i})\right\rangle$$ $$f_1(\vec{p}, \vec{q}, t)=N\int\prod_{i=2}^N d^3\vec{p}_id^3\vec{q}_i\rho(\vec{p}_1=\vec{p}, \vec{q}_1=\vec{q}, \vec{p}_2, \vec{q}_2, \cdots, \vec{p}_N, \vec{q}_N, t)$$ Here, $\langle\mathcal{O}(\vec{p}, \vec{q})\rangle=\int\prod_{i=1}^Nd^3\vec{p}_id^3\vec{q}_i\mathcal{O}(\vec{p}, \vec{q})\rho(\vec{p}, \vec{q})$
Now, here I don't understand what the author exactly means by "the expectation value of finding a particle", I mean how do I quantify "finding a particle"? How is it related to the probability of finding a particle? Also, what does the Dirac delta function in the angle bracket represents? Can someone explain it to me? Thanks for considering my question.
