View source for Fishburn–Shepp inequality

In [[Combinatorics|combinatorial]] mathematics, the '''Fishburn–Shepp inequality''' is an inequality for the number of extensions of partial orders to linear orders, found by {{harvtxt|Fishburn|1984}} and {{harvtxt|Shepp|1982}}.  
It states that if ''x'', ''y'', and ''z'' are incomparable elements of a finite poset, then;-

:<math> P(x<y)P(x<z) \leqslant P((x<y) \wedge (x<z))</math>

where ''P''(*) is the probability that a linear order < extending the partial order has the property *. 

In other words the probability that ''x''&nbsp;<&nbsp;''z'' strictly increases if one adds the condition that ''x''&nbsp;<&nbsp;''y''.  In the language of [[Conditional probability|conditional probability]],

: <math> P(x < z) < P(x < z \mid x < y).</math>

The proof uses the [[Ahlswede–Daykin inequality]].

==References==
*{{Citation |  last1=Fishburn | first1=Peter C. | title=A correlational inequality for linear extensions of a poset | doi=10.1007/BF00565648 | mr=764320  | year=1984 | journal=Order | issn=0167-8094 | volume=1 | issue=2 | pages=127–137}}
*{{springer|id=f/f110080|first=P.C. |last=Fishburn|first2=L.A.|last2= Shepp}}
*{{Citation | doi=10.1214/aop/1176993791 |  last1=Shepp | first1=L. A. | title=The XYZ conjecture and the FKG inequality | jstor=2243391 | mr=659563  | year=1982 | journal=The Annals of Probability | issn=0091-1798 | volume=10 | issue=3 | pages=824–827 | publisher=Institute of Mathematical Statistics| url=https://repository.upenn.edu/cgi/viewcontent.cgi?article=1268&context=statistics_papers | doi-access=free }}

{{DEFAULTSORT:Fishburn-Shepp inequality}}
[[Category:Inequalities]]
[[Category:Combinatorics]]
[[Category:Independence (probability theory)]]

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