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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space. Let $X$ be a [[random variable|0DKRR22AEQ]] with values in $(E_1, \\mathcal{E}_1)$ and $Y$ a random variable with values in $(E_2, \\mathcal{E}_2)$.\n$X$ and $Y$ are independent if for all $A \\in \\mathcal{E}_1$ and $B \\in \\mathcal{E}_2$,\n$$P(X \\in A, \\, Y \\in B) = P(X \\in A) \\, P(Y \\in B).$$\n\n## Remarks\n\nIn other words, the events $\\{X \\in A\\}$ and $\\{Y \\in B\\}$ are [[independent|0P0MZ4X682]] for all $A$ and $B$.\nFor real random variables, it is enough to check $P(X \\le x, Y \\le y) = P(X \\le x) \\, P(Y \\le y)$ for all real $x$ and $y$.\nFor discrete random variables, independence is equivalent to $P(X = x, Y = y) = P(X = x) \\, P(Y = y)$ for all $x$ and $y$.\nIf $X$ and $Y$ are independent and $g$, $h$ are measurable functions, then $g(X)$ and $h(Y)$ are independent.\n\n## Explanations\n\n### L1 human\n\nTwo random variables are independent when knowing the value of one tells you nothing about the other.\nThe results of two separate dice are independent. The result of one die and the sum of both dice are not.\n",
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