Are all variables in a set of random variables independent if all pairs are independent? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:07:26Z http://mathoverflow.net/feeds/question/121190 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121190/are-all-variables-in-a-set-of-random-variables-independent-if-all-pairs-are-indep Are all variables in a set of random variables independent if all pairs are independent? Rorsa 2013-02-08T13:06:22Z 2013-02-08T13:26:42Z <p>If I have a sequence of random variables \$X_1, X_2, \ldots, X_n\$ (possibly infinite) such that all pairwise cdf's are factorized:</p> <p>\$\$F(X_i, X_j) = F_i(X_i) F_j(X_j)\$\$</p> <p>for all pairs \$(X_i, X_j)\$, does it mean that the joint cdf is also factorized? That is:</p> <p>\$\$F(X_1, \ldots, X_n) = \prod_{i=1}^{n} F_i(X_i)\$\$</p> <p>In other words, if I prove that each pair in the sequence is statistically independent of each other, can longer sequences still be non-independent?</p> <p>It seems to me that they can, but I can't come up with a counter example. </p> http://mathoverflow.net/questions/121190/are-all-variables-in-a-set-of-random-variables-independent-if-all-pairs-are-indep/121192#121192 Answer by Steven Landsburg for Are all variables in a set of random variables independent if all pairs are independent? Steven Landsburg 2013-02-08T13:21:11Z 2013-02-08T13:21:11Z <p>The simplest of the many standard counterexamples is when \$(X_1,X_2,X_3)\$ takes the values \$(1,1,1)\$, \$(1,0,0)\$, \$(0,1,0)\$ and \$(0,0,1)\$ all equiprobably.</p> http://mathoverflow.net/questions/121190/are-all-variables-in-a-set-of-random-variables-independent-if-all-pairs-are-indep/121194#121194 Answer by Brendan McKay for Are all variables in a set of random variables independent if all pairs are independent? Brendan McKay 2013-02-08T13:26:42Z 2013-02-08T13:26:42Z <p>Steven's example is indeed the simplest. See chapter 3 of <a href="http://tocs.ulb.tu-darmstadt.de/12587850.pdf" rel="nofollow">this book</a> for counterexamples to lots of similar possibilities.</p>