TITLE: Exponential bulbs?
QUESTION [5 upvotes]: Can the expiration time of a new light bulb be an exponential random variable? (The second problem here namely features such a bulb.)
I'm asking because of the curios memorylessness of the exponential distribution, which I understand like this. With $T$ an exponential variate with parameter $\lambda$ representing time till event $A$ occurs (like time until a bulb burns out), probability $\mathbb{P}\{T > t\} = e^{-\lambda \, t}$ is probability that $A$ doesn't occur till $t$. But if we know that $A$ hasn't occurred till $t$ then probability that $A$ doesn't occur till the additional time $X$ is the same: $\mathbb{P}\{X > t \, | \, T > t\} = e^{-\lambda \, t}$.
Obviously if I leave a new bulb on, check it after a year and, given it still works, I can't really say that the chance of persisting for the same duration stays the same. Can I? A bulb is dying with use, specifically with working time and on-off switching, as stated on the halogen bulb package of mine (2000 operating hours, 50 thousand switchings).
What kind of events and waiting times do follow exponential distribution? I remember my professor mentioning waiting time of supernovas. Why is that exponential?

REPLY [3 votes]: @BruceET answers your question from the physical perspective, in the sense that probability is seen as a property of the physical world out there. However, just so you know, there is also an information theoretic viewpoint of probability (see Probability theory: Logic of Science by E.T. Jaynes, and also his papers).
In this viewpoint probability is a function of information that you have. If you know that the lifetime of bulb is say maximum 2 years, then conditioned on this information you cannot assume an exponential distribution for the lifetime of the bulb. However for someone who doesn't know of the specifications of the bulb but knows only the average rate of failure, exponential distribution is the one to assume. This is because not knowing when failure might occur, he can only assume that it occurs at random times (he has no basis to assume anything different), which gives rise to Poisson distribution for failure times and exponential distribution for lifetimes.