 Solution

Events $(A_n)$ are independent events if for every finite list of distinct indices $i_1,\dots ,i_k$, $\mathbb{P}({\bigcap }_j{A}_{i_j})={\prod }_j\mathbb{P}(A_{i_j})$. 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.