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  clickbait: 'A probability distribution over the collection of joint configurations of all the variables you care about.',
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  text: '[summary: \n$$\\newcommand{\\bR}{\\mathbb{R}}\n\\newcommand{\\bP}{\\mathbb{P}}\n\\newcommand{\\cS}{\\mathcal{S}}\n\\newcommand{\\cF}{\\mathcal{F}}\n\\newcommand{\\gO}{\\Omega}\n\\newcommand{\\go}{\\omega}\n\\newcommand{\\ts}{\\times}$$\n\nA joint probability distribution of real-valued random variables $X_1, X_2, \\cdots, X_n$ is a probability distribution $\\bP$ over $\\bR^n$. The probability of the event $(X_1 \\in A_1, X_2 \\in A_2, \\cdots, X_n \\in A_n)$ is $\\bP(A_1,A_2, \\cdots, A_n)$. \n]\n\n\n$$\n\\newcommand{\\bR}{\\mathbb{R}}\n\\newcommand{\\bP}{\\mathbb{P}}\n\\newcommand{\\cS}{\\mathcal{S}}\n\\newcommand{\\cF}{\\mathcal{F}}\n\\newcommand{\\gO}{\\Omega}\n\\newcommand{\\go}{\\omega}\n\\newcommand{\\ts}{\\times}\n$$\n\n\nA joint probability distribution of real-valued random variables $X_1, X_2, \\cdots, X_n$ is a probability distribution $\\bP$ over $\\bR^n$. The probability of the event $(X_1 \\in A_1, X_2 \\in A_2, \\cdots, X_n \\in A_n)$ is $\\bP(A_1,A_2, \\cdots, A_n)$. \n\n\n# Formal definition\n\nLet $\\{X_i \\}_{i \\in I}$ be a collection of random variables taking values in the spaces $(S_i, \\cS_i)$. Then a joint distribution of the $\\{X_i \\}_{i \\in I}$ is a probability distribution over $\\prod_{i \\in I} S_i$. \n\nIf the $\\{X_i \\}_{i \\in I}$ are defined on a probability space $(\\gO, \\cF, \\bP)$, then $\\bP$ induces a joint distribution of the $X_i$. The induced function is a distribution because an event $(X_1 \\in A_1, X_2 \\in A_2, \\cdots, X_n \\in A_n)$ with $A_k \\in \\cS_k$ can be viewed as the event $\\{ \\go \\in \\gO : X_1(\\go)  \\in A_1, \\cdots, X_n(\\go) \\in A_n\\}$, which is in $\\cF$ because $\\go \\mapsto  (X_1(\\go), \\cdots, X_n(\\go))$ is a measurable map $\\gO \\to S_1 \\ts \\cdots \\ts S_k$.',
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