Problem with the definition of the Empirical Distribution Function Why we have defined the empirical distribution function in an exercise as
$\mathbb{F}_n(x)= \mathbb{P}(Y_n<x)$
More precisely,  why it is strictly inferior?
 A: That is unusual and I think it is a typo because also the usual cdf definition does not use strict inequality. From van de Geer:
The empirical distribution. The unknown $P$ can be estimated from the data in the following way. Suppose first that we are interested in the probability that an observation falls in $A$, where $A$ is a certain set chosen by the researcher. We denote this probability by $P(A)$. Now, from the frequentist point of view, the probability of an event is nothing else than the limit of relative frequencies of occurrences of that event as the number of occasions of possible occurrences $n$ grows without limit. So it is natural to estimate $P(A)$ with the frequency of $A$, i.e, with
$$
\begin{gathered}
P_n(A)=\frac{\text { number of times an observation } X_i \text { falls in } A}{\text { total number of observations }} \\
=\frac{\text { number of } X_i \in A}{n} .
\end{gathered}
$$
We now define the empirical measure $P_n$ as the probability law that assigns to a set $A$ the probability $P_n(A)$. We regard $P_n$ as an estimator of the unknown $P$.
The empirical distribution function. The distribution function of $X$ is defined as
$$
F(x)=P(X \leq x),
$$
and the empirical distribution function is
$$
\hat{F}_n(x)=\frac{\text { number of } X_i \leq x}{n} .
$$

