As defined in the text of CMU statistics notes. The Hoeffding's inequality is defined as: $$P(|\bar{X}-\mu|\geq t)\leq 2 \exp\left(\frac{2n^2t^2}{\sum_{i=1}^n{(b_i -a_i)^2}}\right)$$ where $\mu=E[X_i]$.
I understand that in machine learning terms, as in PAC analysis, $\bar{X}-\mu$ means the difference between the observed mean and the true mean of the random variable $X$. However, what confuses me is the choice of the represnetation of $E[X_i]$, an expectation of a member $X_i\in X$ instead of $E[\bar{X}]$.
Is $E[X_i]$ a conventional expression for $E[\bar{X}]$? If yes, in what sense it is correct to put it this way?
By writing $\mu=E(X_i)$ we mean that all the random variables $X_1,...,X_n$ has the same mean $\mu$ and hence $\overline{X}$ also has mean $\mu$.