  For a random sample of points in $\mathbb{R}$, we consider the number of
pairs whose members are nearest neighbors (NN) to each other and the number of
pairs sharing a common NN. The first type of pairs are called reflexive NNs
whereas latter type of pairs are called shared NNs. In this article, we
consider the case where the random sample of size $n$ is from the uniform
distribution on an interval. We denote the number of reflexive NN pairs and the
number of shared NN pairs in the sample as $R_n$ and $Q_n$, respectively. We
derive the exact forms of the expected value and the variance for both $R_n$
and $Q_n$, and derive a recurrence relation for $R_n$ which may also be used to
compute the exact probability mass function of $R_n$. Our approach is a novel
method for finding the pmf of $R_n$ and agrees with the results in literature.
We also present SLLN and CLT results for both $R_n$ and $Q_n$ as $n$ goes to
infinity.
