I'm trying to learn about negative association of random variables. A definition can be found here: http://www.cs.cmu.edu/~dwajc/notes/Negative%20Association.pdf.

Now, consider the following question:

Let $X_1$, $X_2$ be independent but not necessarily identically distributed random variables. Let $σ_1, σ_2$ be a random permutation on $\{1, 2\}$. Are the variables $Y_i = X_{σ_i}$ negatively associated? Why?

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    $\begingroup$ Seems like for each fixed outcome of $X_1,X_2$, the $Y_i$ are negatively associated (taking probability only over $\sigma$) by Lemma 8. But I'm not sure it's easy to continue from there. $\endgroup$ – usul Sep 21 '19 at 2:34

The answer is no. Why? Suppose $X_1=2$ a.s. and $X_2$ takes the values 1 and 3 with Probability $1/2$ each. Let $f$ be the indicator of $[3,\infty)$ and let $g$ be the indicator of $[2,\infty)$. Then the expectation of $f(Y_1) g(Y_2)$ is $1/4$. But $f(Y_1)$ has expectation $1/4$ and $g(Y_2)$ has expectation $3/4$, so negative association fails.


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