Suppose that $X_1,\dots,X_n$ are i.i.d. random variables having a CDF (cumulative distribution function) $F$. For each fixed $x$, I am asked to determine the maximum likelihood estimate of $F(x)$.
I am having difficulty understanding what the question is asking. For one, I do not see how to recover a probability density (or mass) function from $F$, let alone how to maximize the likelihood function without a given parameter space.
If anyone could shed some light on this it would be greatly appreciated.
I now understand that I should take the MLE will be the empirical distribution. However, I am still having difficulty proving this directly from the definition.