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

(i)	$X_j$ is a D202: Random variable on $P$ for each $j \in J$

Then $X = \{ X_j \}_{j \in J}$ is an independent random collection on $P$ if and only if \begin{equation} \forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \text{ distinct } j_1, \ldots, j_N \in J \left[ \{ X_{j_1} \in E_{j_1} \}, \ldots, \{ X_{j_N} \in E_{j_N} \} \in \mathcal{F} \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N \{ X_{j_n} \in E_{j_n} \} \right) = \prod_{n = 1}^N \mathbb{P}(X_{j_n} \in E_{j_n}) \right] \end{equation}