  Consider the distances $\tilde{R}_o$ and $R_o$ from the nucleus to a
uniformly random point in the 0-cell and the typical cell, respectively, of the
$d$-dimensional Poisson-Voronoi (PV) tessellation. The main objective of this
paper is to characterize the exact distributions of $\tilde{R}_o$ and $R_o$.
First, using the well-known relationship between the 0-cell and the typical
cell, we show that the random variable $\tilde{R}_o$ is equivalent in
distribution to the contact distance of the Poisson point process. Next, we
derive a multi-integral expression for the exact distribution of $R_o$.
Further, we derive a closed-form approximate expression for the distribution of
$R_o$, which is the contact distribution with a mean corrected by a factor
equal to the ratio of the mean volumes of the 0-cell and the typical cell. An
additional outcome of our analysis is a direct proof of the well-known
spherical property of the PV cells having a large inball.
