An important quantity in a statistical analysis is the so-called covariance.

Consider the set \( \{X_i\} \) of \( n \) stochastic variables (not necessarily uncorrelated) with the multivariate PDF \( P(x_1,\dots,x_n) \). The covariance of two of the stochastic variables, \( X_i \) and \( X_j \), is defined as follows $$ \begin{align} \mathrm{Cov}(X_i,\,X_j) & = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle \tag{6}\\ &=\int\cdots\int (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)P(x_1,\dots,x_n)\,dx_1\dots dx_n, \tag{7} \end{align} $$ with $$ \begin{equation*} \langle x_i\rangle = \int\cdots\int x_i P(x_1,\dots,x_n)\,dx_1\dots dx_n. \end{equation*} $$