= Solution
For a statistically homogeneous matter distribution, the <cosmological two-point correlation function> of the mean-zero <density contrast> $\delta=(\rho-\bar\rho)/\bar\rho$ is
$$
\boxed{\xi(r)=\langle\delta(\mathbf x)\delta(\mathbf x+\mathbf r)\rangle,
\qquad r=|\mathbf r|.}
$$
<Spatial homogeneity> removes dependence on $\mathbf x$, and <isotropy> reduces the separation argument to its magnitude. For discrete <galaxies>, the <galaxy two-point correlation function> equivalently specifies the excess pair probability at distinct positions:
$$
dP_{12}=\bar n^2[1+\xi(r)]\,dV_1dV_2.
$$
Consequently the mean density of other <galaxies>, conditioned on a <galaxy> at the origin, is $\bar n[1+\xi(r)]$. The self-pair at zero separation is excluded from this definition; its <Dirac delta> contribution is the <galaxy power spectrum shot noise> of the unsmoothed number-density field.
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