# Negatively Correlated Events

I showed the following inequality to a colleague, where $A$ and the $B_i$ are all events: $$\Pr\left(A \mid \bigwedge_{i = 1}^n \overline{B_i} \right) \leq \Pr(A)$$ He summarized, "So $A$ is negatively correlated with the $B_i$."

I have never heard this phrase used to refer to events. Even when it is used with respect to random variables, I'm not sure it is quite the same thing.

Could a more experience probabilist chime in? Does his terminology make sense? Is there a better word?

-

## 1 Answer

Does his terminology make sense?

It does: events $A$ and $B$ are negatively correlated if and only if $\mathbb P(A\cap B)\leqslant\mathbb P(A)\mathbb P(B)$, random variables $X$ and $Y$ are negatively correlated if and only if $\mathbb E(XY)\leqslant\mathbb E(X)\mathbb E(Y)$.

-
As surely your colleague would have told you on the spot if only you had asked them. –  Did Dec 11 '12 at 7:12
I just wanted a second opinion. Thanks. –  Austin Mohr Dec 11 '12 at 9:47
I just noticed that I slightly mistyped the inequality by neglecting the complementation. Your answer is still good, but my colleague probably should have said that $A$ is negatively correlated with the complement of the $B_i$s. –  Austin Mohr Dec 21 '12 at 18:50
It's worth pointing out that correlation of events is a special case of correlation of variables: just take the indicator functions of the events. –  Chris Eagle Dec 21 '12 at 18:52
AustinMohr: You should have left the question as it was, the modification only adds confusion. –  Did Dec 21 '12 at 21:44