# Simple question regarding negatively associated random variables

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?

• 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. – 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.