Disclaimer: I don't know much about statistics
$X\in \mathbb{R}^{N \times n} $ or $ X:\{1, \ldots, N\} \rightarrow \mathbb{R}^{n} $ describe a measurement , where $N$ is the number of objects examined and each object has a data set $x_j=X(j) \in \mathbb{R}^{n}$ with $ j\in \{1, \ldots, N\}$ is assigned. For $j\in \{1, \ldots, N\}, X_j \in \mathbb{R}^{N}$ describes the $j-$th component of the measurement. Why is the following true
$$E\left(X_n X_{k}\right)-(E X_n) \cdot \left(E X_{k}\right)=\operatorname{Cov}\left(X_n,X_k \right) \\ ?$$
I only know this formula of the covariance: $\operatorname{Cov}(X, Y):=E((X-E X) \cdot(Y-E Y)) \in \mathbb{R} $, where $(X-E X)$ and $ (Y-E Y)$ are the centered values of $X$ and $Y$ respectively.
Just multiply out the expression and consider $E(a*X+b)=a*E(X)+b$, here e.g. $E(X*E(Y))=E(Y)*E(X)$, taking $E(Y)$ as $a$