Question

Prove that x-s-t is a factor of x^3 - s^3 -t^3 -3st(s+t)

Answer 1

1. Introduction. This paper discusses a special form of positive dependence. Positive dependence may refer to two random variables that have a positive covariance, but other definitions of positive dependence have been proposed as well; see [24] for an overview. Random variables X = (X1, . . . , Xd) are said to be associated if cov{f(X), g(X)} ≥ 0 for any two non-decreasing functions f and g for which E|f(X)|, E|g(X)|, and E|f(X)g(X)| all exist [13]. This notion has important applications in probability theory and statistical physics; see, for example, [28, 29]. However, association may be difficult to verify in a specific context. The celebrated FKG theorem, formulated by Fortuin, Kasteleyn, and Ginibre in [14], introduces an alternative notion and establishes that X are associated if ∗ SF was supported in part by an NSERC Discovery Research Grant, KS by grant #FA9550-12-1-0392 from the U.S. Air Force Office of Scientific Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA), CU by the Austrian Science Fund (FWF) Y 903-N35, and PZ by the European Union Seventh Framework Programme PIOF-GA-2011-300975. MSC 2010 subject classifications: Primary 60E15, 62H99; secondary 15B48 Keywords and phrases: Association, concentration graph, conditional Gaussian distribution, faithfulness, graphical models, log-linear interactions, Markov property, positive