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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space and let $I$ be an index set. For each $i \\in I$, let $X_i$ be a [[random variable|0DKRR22AEQ]] with values in $(E_i, \\mathcal{E}_i)$.\nThe family $(X_i)_{i \\in I}$ is independent if for each finite nonempty set $J \\subseteq I$ and each choice of $B_j \\in \\mathcal{E}_j$ for $j \\in J$,\n$$P\\left(\\bigcap_{j \\in J} \\{X_j \\in B_j\\}\\right) = \\prod_{j \\in J} P(X_j \\in B_j).$$\n\n## Remarks\n\nEquivalently, the σ-algebras $\\sigma(X_i) = \\{X_i^{-1}(B) : B \\in \\mathcal{E}_i\\}$ are [[independent|09T8ZHST1V]].\nFor two random variables this is the [[independence of two random variables|0E9502NT92]].\nMutual independence implies pairwise independence, but the converse is false, as for [[events|0AR4B3K5QD]].\nFor real random variables $X_1, \\dots, X_n$, independence holds if and only if the joint cumulative distribution function is the product of the marginal ones.\n\n## Explanations\n\n### L1 human\n\nIndependent random variables come from separate parts of an experiment that do not influence each other.\nFor $n$ tosses of a coin, the results of the tosses are independent random variables.\n",
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