In Cosmology, we have the co-moving distance (assuming $\Omega_k=0$), $$D_C=\frac c{H_0}\int_0^z\frac{dz'}{\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}}$$ and we also have the total co-moving volume formula $$V=\frac{4\pi}3 D_C^3$$Then we can use what is called the $\langle V/V_{max}\rangle$ test to test if a sample of objects has uniform co-moving density & luminosity that is constant in time. The number of objects per unit co-moving volume with luminosity in the range $(L,L+dL)$ is given by $\Phi(L)$. Then the total number of objects in the sample is $$\int_0^\infty\Phi(L)\int_0^{V_{max}(L)}\mathrm dV\,\mathrm dL$$ If we have a uniform distribution of objects, then the value of this expectation should be $1/2$, as described by this similar question asked on Student Room, which did not get any responses.

Question: From these definitions, I don't know how to get a value for $\langle V/V_{max}\rangle$, nor do I know what the explicit formula is. What is the explicit formula for $\langle V/V_{max}\rangle$? Is it that integral? Clearly, $V$ depends only on the value of $z$, but I don't know what a uniform distribution of objects implies about the distribution of $z$. Can someone help me understand this a bit better?