Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X_1, \, \dots, \, X_N : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$
(ii)	$\mathbb{E}|X_1|^2, \, \dots, \, \mathbb{E} |X_N|^2 < \infty$

Then $X_1, \, \dots, \, X_N$ is an D3842: Uncorrelated random collection on $P$ if and only if \begin{equation} \mathbb{E} \left( \prod_{n = 1}^N X_n \right) = \prod_{n = 1}^N \mathbb{E}(X_n) \end{equation}