updating a distribution based on the non-occurence of an event I have a dataset of devices, and the delay (days) in the time from when they are turned on to when they first register on the network.  I've modeled it as a Gamma distribution, which decays very quickly.
Let's suppose that I have a sample of new devices, which I know have not registered themselves within 48 hours.  How do I update the distribution to take into account this information?  I'd like to be able to provide a conditional probability that the devices will be registered on day 3, day 4, etc - given that they weren't registered within the first two days.
If I just use the existing distribution, I'd still have probabilities at day 1, and day 2 - even though those times have passed and are no longer valid.  But at the same time, I can't just shift the distribution forward 48 hours, as the probability for day 2 would be too high.
Should I take a Bayesian approach, or is there some type of distribution which is suitable for these types of calculations?
 A: You could take a Bayesian approach, as you say, and if your prior is a Gamma then your posterior will also be a Gamma, although you should make sure that the variable you are measuring is indeed continuous, otherwise your very first step (and subsequently everything that follows) will be incorrect.
What you are asking for in this case however could be answered straightforwardly with the probability $\mathbb{P}[X \leq s\mid X > t], s > t$. The detailed calculation of this will depend on whether your random variable is discrete or continuous, but in general this is equal to $$\dfrac{\mathbb{P}[t < X\leq s]}{\mathbb{P}[X>t]} = \dfrac{F(s)-F(t)}{1-F(t)}.$$
In the discrete case, you can specify exactly at which point you want the probability to be evaluated, i.e. the event $\{X = s\}$, but be careful with the cumulative probabilities, as in the discrete case there is a difference between $\{X < s\}$ and $\{X \leq s\}$.
That being said, after your initial fitting of the distribution (on the "original" sample), you can use the estimated distribution to make predictions, so to speak, about devices that have not yet been tested, especially since you are interested in the case where the connection doesn't happen right away. If the probabilities you get for certain events seem to match the actual data from the new devices, your distributional fit seems to perform well. Otherwise, you either need to recalibrate the parameters by including the new devices' times in your data, or think of a different distributional model altogether.
In any case, checking whether your model needs an update from time to time is never a bad idea.
