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Let's assume we have a physical experiment which runs continuously and shows some events we are interested in.

An event could be detecting a certain type of particle.
Or, to make it more demonstrative, the event may be overheating of our contraption. (We know it will automatically recover.)

We start our experiment, knowing that it should not be able to overheat at all.

Let's consider various possible observations after running it for one day:

We started with the assumption that it does not overheat at all.

  • It just runs smoothly, and it never overheated.
    • That tells us we were probably right.
  • It overheats at random times about once per minute.
    • We now expect the next overheat event to occur in roughly a minute
  • It overheated once.
    • What do we know based on this?

We learned that it can overheat. Should we assume that it will overheat again? The probability of overheating is not 0 as we expected. Anything else?

One may argue the method of statistics just does not apply here, because we do not have a set of samples to base on. But there seems to be some information we get.

Do we just update our estimate when an event occurs to be the number of events up to now per time passed since start? Could we say anything about how sure we are about it?

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  • $\begingroup$ You cannot say how sure you are about the chance of overheating. You have a standard deviation of 0 and cannot construct a confidence interval. In this case, I believe you are limited to saying the event happened and given that you know this was a true event (the sensor was working correctly and so on), it may happen again but you do not know how often and cannot even say how confident you are it will happen again. $\endgroup$
    – M Waz
    Jul 17, 2019 at 15:57
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    $\begingroup$ There is a rich literature on this called reliability analysis. The trick is that a "single event" is comprised of many measures: such as the time series of inputs given to the system, any measures of system state (such as operating temperature prior to over heat failure), the actually timing of failure, and the extent of failure (did the system recover? Fully? how long did it take to recover? ) $\endgroup$
    – AdamO
    Jul 17, 2019 at 16:15
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    $\begingroup$ @MWaz You can actually maximize a (single parameter) likelihood from a single observation, like an exponential distribution for time-to-event. Due to the mean-variance relationship, you get a standard error as well. $\endgroup$
    – AdamO
    Jul 17, 2019 at 16:17

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Granted that a single observation may not tell you everything you'd like to know, but continuing from @AdamO's Comment, information from one occurrence may be useful.

Now you know that overheating can happen. You might assume that such events happen according to a Poisson process with rate $\lambda$ so that times between events are exponential with rate $\lambda$ and mean $\mu = 1/\lambda.$

If $X \sim \mathsf{Exp}(\text{rate} =\lambda),$ then $X/\mu \sim \mathsf{Exp}(1).$ Consequently,

$$0.95 = P(L \le X/\mu \le U) = P(X/U \le \mu \le X/L),$$ where $L$ and $U$ cut probability 0.025 from the lower and upper tails, respectively, of $\mathsf{Exp}(1).$

If $X = 2.3$ hours, a computation in R gives the 95% CI $(.62,\, 90.8)$ hours. Not as much as you might like to know, nevertheless some information. (Without intervention, you shouldn't be astonished if there's another overheating event within the week.)

2.3/qexp(c(.975,.025), 1)
[1]  0.6234956 90.8451475
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