Failure time distributions from an uncertain starting point

Question

I have a data analysis problem in the laboratory which I think can be reframed as a kind of reliability question.

Question: Suppose a warehouse is filled with light bulbs. The probability distribution of lifetimes of the light bulbs is $P(t)$. Now consider that the light bulbs have unknown ages at time $t=0$ (i.e., they were replaced some time ago, but we can't know exactly when). How long should we expect we have until a light bulb fails?

Attempt: The lifetime of a lightbulb is $t + t'$, where $t$ is the time since we started observing the bulb and $t'$ is the unknown time between installation and the period when we started observing the bulb.

If failure occurs at $t+t'$, we have $\tau = t+t'$ distributed as $P(\tau)$. However, we can only measure the distribution of $t$, i.e. the lifetimes since we started paying attention: call that $O(t)$. Given that we don't know the unknown time that lapsed before we started watching the bulbs, a reflection of our ignorance is to assume this time is a uniform distribution. Since the time $\tau$ is the sum of two random variables, we can conclude $$ P(\tau) = \lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T O(\tau-t) dt.$$

I do not think this is correct since the result diverges if $O(\tau-t)$ is an exponential distribution. I believe the problem is that I assumed $t$ and $t'$, the observed and un-observed times, were independent.

Is this a known problem in reliability theory? Do I have enough information to estimate the probability of lifetimes of an array of machines without information about when the machines were first turned on?

Answer 1

I assumed $t$ and $t′$, the observed and un-observed times, were independent.

let $t'$ = The unknown age achieved without failure.

let $\Delta$ = additional hours to operate.

• if you assume your failure distribution is uniform then, time is irrelevant always. (No matter what the values of times are, probability of failure is always same which is unrealistic.) $t'$ and $\Delta$ are irrelevant for this case.
• if you assume your failure distribution is exponential then, the exponential distribution is memoryless because the past has no bearing on its future behavior. This means that probability of failure from $t1=0$ to $t2=\Delta+t1$ is same with probability of failure from $t3>0$ to $t4=\Delta+t3$. The exponential distribution only cares for $\Delta$ that is duration without any aging effect. So again you don't need to concern for the unknown $t'$ value.

Do I have enough information to estimate the probability of lifetimes of an array of machines without information about when the machines were first turned on?

you should search for distribution fitting/parameter estimation with right censored data.

By the way, if a machine of yours is a repairable system in contrast of a light bulb (a remove/replace item) then take extra caution if you'll take some serious real life decisions.

... I believe the problem is that I assumed t and t′, the observed and un-observed times, were independent.

I think you're right. You can't assume $t$ and $t′$ are independent in all boundary of reliability theory. Statistical analyses based on $t$ and $t′$ are mostly dependent. You may consider to search for conditional reliability function to arrive to conditional unreliability function (or conditional cumulative distribution function) if you are unaware of.

A recommended book if you have access: Probability Distributions Used in Reliability Engineering; Andrew N. O’Connor