I've got a sensor network - a collection of hipster detectors planted at various locations in Brooklyn. Due to power limitations, the sensors are not connected to any network on a regular basis. Rather than immediately transmitting information to the central data collector, the sensors instead wake up at random times and ping the network at some later time. I.e., there is a delayed reaction between the event occurring and the information of the event being transmitted.

I've got a website selling real estate. There is a long lead time between a visitor arriving on my website and actually purchasing a home. I had a large burst of traffic 10 minutes ago, and in the past 10 minutes not a single one of those visitors has purchased a home. Is this evidence that my conversion rate is low? Of course not - there is again a delayed reaction between a site visit and a conversion.

The problem of delayed reactions is a fairly general problem that comes up in a variety of cases. I've run into this issue in conversion rate optimization, sensor networks, anomaly detection and several more. How can we take such issues into account statistically?

## The problem statement

This is a case where the same math can be used to describe two different problems.

### Sensor Networks

In the sensor network formulation, we can state the problem as follows.

We have a sequence of events, $e_1, \ldots, e_N$ which occur at times $t_{e,1}, \ldots, t_{e,N}$. If we wait long enough, any event $e_i$ has a probability $\gamma$ of being observed.

If the observation occurs, it will happen at some time $t_{o,1}, \ldots, t_{o,M}$, for some $M < N$ since not all events will be observed. Critically, we will assume that the probability of observation before a fixed time is solely a function of the delay $t_i = t_{o,i} - t_{e,i}$. As an example, suppose event 1 occurred at 1PM and event 2 occurred at 2PM. Then the probability of detecting event 1 before 5PM (1PM + 4 hours) is the same as the probability of detecting event 2 before 6PM (2PM + 4 hours).