Singularity Notes

In one of our projects, we encountered this dilemma where we had to nitpick on (the probability of) co-occurrence of a pair of events and correlation between the pair of events. Here is my attempt at disambiguating between the two. Looking forward to any pokes at loopholes in my argument. Consider two events e1 and e2 that have a temporal signature. For instance, they could be login events of two users on a computer system across time. Let us also assume that time is organized as discrete units of constant duration each (say one hour). We want to now compare the login behaviour of e1 and e2 over time. We need to find out whether e1 and e2 are taking place independently or are they correlated. Do they tend to occur together (i.e. co-occur) or do they take place independent of one another? This is where terminologies are freely used and things start getting a bit confusing. So to clear the confusion, we need to define our terms more precisely. Co-occurrence is simply the probability that