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?



Simple question regarding negatively associated random variables
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. 


