# Component reliability with an uncertain starting time

I have a lab data analysis challenge which I think can be viewed as a kind of reliability analysis problem.

Imagine a warehouse is filled with light bulbs that were installed at some unknown times in the past. These have operated up to the present time from whenever they were installed. Now at some time $$t=0$$ we start recording the failure times of bulbs in the warehouse. After many of the bulbs burn out, we find the statistical distribution of the failure times $$P(t)$$.

This $$P(t)$$ obviously does not represent the actual lifetime distribution of the bulbs. It is only a proxy for it, and its mean will only be a lower bound for the actual mean lifetime. But assuming that the installation times of the bulbs are randomized, is it possible to infer the actual lifetime distribution $$Q(t)$$ of the bulbs?

Relating to this I wonder (1) is there a relationship between $$P(t)$$ and $$Q(t)$$, and (2) is this a common problem in reliability analysis? Any key words I might search to find references on this topic would be helpful.

Some thoughts: If the distribution of installation times (i.e., the length of time in the past from $$t=0$$ when each bulb was installed) is $$U(\tau)$$, while the censored lifetimes $$t$$ we measure are distributed as $$P(t)$$, we should have $$t' = t- \tau$$ as the real lifetimes distributed as $$Q(t')$$. Therefore if the installation times are independent of the lifetimes, we have that the distribution $$Q(t')$$ is the distribution of the difference of $$t$$ and $$\tau$$: $$Q(t') = \int_0^\infty \int_0^\infty U(\tau)P(t)\delta(t'-[t-\tau])d\tau dt.$$ But what is $$U$$ ? And shouldn't it be related to $$Q(t)$$?