Suppose you have the logs of a web server. In these logs you have tuples of this kind:
user1, timestamp1
user1, timestamp2
user1, timestamp3
user2, timestamp4
user1, timestamp5
...
These timestamps represent e.g. users' clicks. Now, user1
will visit the site multiple times (sessions) during the month, and you'll have bursts of clicks from each user during each session (supposing that when a user visits your site, he'll click on multiple pages).
Suppose you want to partition these burst of clicks in the sessions that generated them, but you don't have any additional source of information, only the list of timestamps. If you compute the distribution of intervals between two consequent clicks from the same user, you will obtain a long-tailed distribution. Intuitively, you'd look for a "cut parameter", e.g. N seconds, where if timestamp_{i+1} - timestamp{i} > N
, then your timestamp_{i+1}
is the beginning of the new session.
The problem is that this distribution in reality is a mixture of two variables: X = "interval between two consequent clicks in the same session" and Y = "interval between the last click of the previous session and the first of the new one".
The question is, how to estimate this N, that divides the two distributions (with a bit of overlap, possibly) just by looking at the burst of clicks?