# Exponential bulbs?

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?

## 2 Answers

Extended Comment:

Your criticism of the exponential distribution as a model for lifetimes of incandescent light bulbs is appropriate. (Eventually tungsten will 'boil away' from the filament and the bulb will die.)

A device has an exponentially distributed lifetime if it dies due to random accidents and wearing out is not an issue. Certain computer chips may have exponential lifetimes. At least in meaningful usage they die by random hits from cosmic rays, random exposure to electric shock, accidental overheating, and so on. They are 'complicated pieces of sand' and there is nothing to wear out. (On a cosmic time scale, one could talk about chemical or particle decay, but that is not relevant in common applications.)

Certain devices that are continually maintained in essentially-new condition might also have exponential lifetimes. A very meticulously pampered auto might have almost the same additional lifetime as if it were new. (But there comes a point where there are too many short lived rubber and plastic parts to replace.)

The exponential distribution is probably used too often in applications. First, because its no-memory property means that one does not have to take past history into account; second, because the math of the exponential distribution is so easy.

If a used item has no better (or worse) life expectancy than a new one, then it is a candidate for an exponential lifetime model. Weibull models are generally messier than exponential special cases, but they can model some situations in which 'used is worse than new' (because of wear) or 'used is better than new' (having survived burn-in).

A major distinction between "reliability theory" and "survival analysis" is that reliability theory is largely used for physical systems where exponential models are often reasonable, while survival analysis is largely used for biological organisms (including humans) that die by a combination of accidents and deterioration with age.

@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.