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  "body": "\n## Statement\n\nLet $(P, \\le)$ be a finite [[partially ordered set|0QV2JS9RJ3]] and let $x, y, z \\in P$.\nChoose a [[linear extension|W8DZ5KGGHN]] $\\prec$ of $P$ uniformly at random among all linear extensions, and write $\\Pr$ for the resulting probability.\nThen\n$$\\Pr(x \\prec y \\text{ and } x \\prec z) \\ge \\Pr(x \\prec y) \\, \\Pr(x \\prec z).$$\n\n## Remarks\n\nShepp proved the inequality using the [[FKG inequality|DCXJGS6E6D]] on a suitable distributive lattice of integer labellings of $P$.\nIt says that the events \"$x$ comes before $y$\" and \"$x$ comes before $z$\" are positively correlated.\nThis matches intuition: if $x$ comes early relative to $y$, it tends to come early in general.\n\nExample: let $P$ be the antichain on $\\set{x, y, z}$.\nAll $6$ orderings are linear extensions, so $\\Pr(x \\prec y) = \\Pr(x \\prec z) = 1/2$, and $x$ comes first in $2$ of the $6$ orderings, so the left side is $1/3 \\ge 1/4$.\n\n## Explanations\n\n### L1 human\n\nPut the elements of a poset in a random order that respects the poset.\nKnowing that $x$ comes before $y$ makes it more likely, or at least not less likely, that $x$ also comes before $z$.\nThe XYZ inequality states this in numbers.\n\n### L2 human\n\nThe inequality belongs to the family of correlation inequalities for linear extensions, studied in connection with sorting and with the 1/3–2/3 conjecture.\nIts proof is a model use of the FKG inequality.\nOne looks at order-preserving labellings of $P$ by a large chain, which form a distributive lattice.\nOne applies FKG there and then lets the chain grow.\n",
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