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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space, let $I$ be an index set, and for each $i \\in I$ let $\\mathcal{F}_i \\subseteq \\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $\\Omega$.\nThe family $(\\mathcal{F}_i)_{i \\in I}$ is independent if for each finite nonempty subset $J \\subseteq I$ and each choice of events $A_j \\in \\mathcal{F}_j$ for $j \\in J$,\n$$P\\left(\\bigcap_{j \\in J} A_j\\right) = \\prod_{j \\in J} P(A_j).$$\n\n## Remarks\n\nEvents $(A_i)_{i \\in I}$ are [[mutually independent|0GCVEV6BJA]] if and only if the σ-algebras $\\{\\varnothing, A_i, A_i^c, \\Omega\\}$ are independent.\nRandom variables are independent when the σ-algebras that they generate are independent.\nTo check independence, it is enough to check the product formula on a generating family of each $\\mathcal{F}_i$ that is closed under finite intersections. This follows from Dynkin's π-λ theorem.\n\n## Explanations\n\n### L1 human\n\nA σ-algebra holds all the information of one part of an experiment, for example everything about the first roll of a die.\nTwo such σ-algebras are independent when information about one part never changes the chances about the other part.\n",
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