I need to prove that:
a) sample median is a plug-in estimator
b) sample quantile is a plug-in estimator,
but I have no idea where to start;
In a) :
$M = \begin{cases} X_{\left(\frac{n+1}{2}:n\right)} & \text{if } n \text{ is odd} \\[6pt] \frac{X_{\left(\frac{n}{2}:n\right)} + X_{\left(\frac{n}{2}+1:n\right)} }{2} & \text{if } n \text{ is even} \end{cases} $
but I don't know how to connect this with empirical CDF