  In this paper, we prove a local limit theorem and a refined continuity
correction for the negative binomial distribution. We present two applications
of the results. First, we find the asymptotics of the median for a Negative
Binomial (r,p) random variable jittered by a Uniform (0,1), which answers a
problem left open in Coeurjolly \& Tr\'epanier (2020). This is used to
construct a simple, robust and consistent estimator of the parameter $p$, when
$r > 0$ is known. Second, we find an upper bound on the Le Cam distance between
negative binomial and normal experiments.
