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How would you express the inequalities $P(A\cap B)\lessgtr P(A)P(B)$?

If two random variables are correlated, $E[XY]\lessgtr E[X]E[Y]$, we call them positively or negatively correlated according to the direction of the inequality. Positively correlated variables tend to vary in the same direction, and negatively correlated variables tend to vary in opposite directions.

I just realized that I don't really know how to express the analogue for dependent events – the fact that the events are more or less likely to occur together than one might expect from their individual probabilities. I could say that their indicator variables are positively or negatively correlated, and it seems to be a common abuse of terminology to say that the events themselves are positively or negatively correlated, but that seems suboptimal, since without the adverb indicating the sign we'd call them dependent and not correlated.

Is there such a thing as positively or negatively dependent events? A Google search for "negatively dependent" mostly turns up definitions of "negatively dependent random variables" (e.g. here and here), not negatively dependent events. How else could this concept be expressed? Is "positively or negatively correlated" the least bad option?

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If $A,B$ are events, you say $B$ attracts $A$ if $P(A\mid B) \gt P(A)$. You say $B$ repels $A$ if $P(A\mid B) \lt P(A)$.

So, in fact, $$B \text{ attracts } A\iff P(A\cap B) \gt P(A)P(B) \iff A \text{ attracts } B$$

and

$$B \text{ repels } A\iff P(A\cap B) \lt P(A)P(B) \iff A \text{ repels } B.$$

Here are some references but the terms don't appear to have widespread use:

Introduction to Probability, Grinstead and Snell, pg. 160

Random Phenomena: Fundamentals of Probability and Statistics for Engineers, Ogunnaike, pg. 83

Elementary Probability, Stirzaker, pg. 56

Probability and Random Processes, Grimmett and Strizaker, pg. 24

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  • $\begingroup$ Wow, thanks, that's an attractive solution! :-) $\endgroup$ – joriki Sep 12 '15 at 0:25
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If we say two variables are correlated and don't attach the adverb we have the same problem - the variables are dependent and we don't know the particular manner. When it's important we qualify it with 'positively/negatively'.

When we speak of correlation of events the custom of omitting "indicator function of" is common and, I think, harmless. So wording $P(A\cap B) < P(A)P(B)$ as '$A$ and $B$ are negatively correlated (events)' isn't clouding anything and, just as in the general case, requires mentioning the manner of correlation.

TL;DR: Your least bad option may even be a good option.

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    $\begingroup$ The first sentence doesn't correspond to the problem I was trying to describe, which isn't that we don't know the particular manner if we don't attach an adverb for the sign. My problem is that variables are correlated and events are dependent, so when we call events positively or negatively correlated we're appropriating an adjective that's not applied to events without the sign adverb. This problem doesn't occur for variables because positively or negatively correlated variables are in fact correlated variables. Whereas positively or negatively correlated events aren't correlated events. $\endgroup$ – joriki Sep 11 '15 at 21:09
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    $\begingroup$ I see. Well, I think it is common practice to conflate the events with their indicator random variables, the abuse you mentioned. My take is that the idea of signed dependence can be phrased as "$\pm$ly correlated events" and that, should you deign to do so in your own writing you can expect to be understood. $\endgroup$ – Titus Sep 11 '15 at 21:32
  • $\begingroup$ It would be good to have a few more readers weigh in though; maybe there is specific vocabulary for this in regions of the literature I haven't dipped into. $\endgroup$ – Titus Sep 11 '15 at 21:34

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