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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

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