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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $(E, \\mathcal{E})$ be a set $E$ with a [[σ-algebra|04WSW8T3SY]] $\\mathcal{E}$.\nA random variable with values in $E$ is a [[function|0XHDTE5YMZ]] $X \\colon \\Omega \\to E$ such that the [[preimage|0FGCFY1AZX]] $X^{-1}(B) = \\{\\omega \\in \\Omega : X(\\omega) \\in B\\}$ is in $\\mathcal{F}$ for each $B \\in \\mathcal{E}$.\nA real random variable is a random variable with values in $\\mathbb{R}$, with the [[Borel σ-algebra|0BVHZXWKCD]] on $\\mathbb{R}$.\n\n## Remarks\n\nThe condition says that $X$ is a measurable function. It makes $\\{X \\in B\\}$ an event, so $P(X \\in B)$ is defined.\nA real function $X$ on $\\Omega$ is a random variable if and only if $\\{X \\le x\\} \\in \\mathcal{F}$ for each $x \\in \\mathbb{R}$.\nFor two rolls of a die, $\\Omega = \\{1, \\dots, 6\\}^2$ and the sum $X(a, b) = a + b$ is a random variable.\nSums, products and continuous functions of real random variables are random variables.\n\n## Explanations\n\n### L1 human\n\nA random variable is a number that you read off from the result of the experiment.\nFor two rolls of a die, the sum of the two numbers is a random variable.\nIt is $7$ for the result $(3, 4)$ and $12$ for the result $(6, 6)$.\n",
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