= Solution
Events $(A_n)$ are <independent events> if for every finite list of distinct indices $i_1,\ldots,i_k$, $\mathbb P(\bigcap_jA_{i_j})=\prod_j\mathbb P(A_{i_j})$. Random variables are <independent random variables> if for every such list and <Borel sets> $B_j\subseteq\mathbb R$, $\mathbb P(X_{i_j}\in B_j\text{ for all }j)=\prod_j\mathbb P(X_{i_j}\in B_j)$. This is equivalent to independence of their generated <sigma-algebras>; <pairwise independence> alone is insufficient.
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