# Do any electric lights have statistical lifetimes X~Exp(λ)?

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## Main Question or Discussion Point

I've come across a number of problems in elementary probability theory and statistics that can be exemplified as follows:
The lifetime, in years, of a certain class of light bulbs has an exponential distribution with parameter λ = 2. What is the probability that a bulb selected at random from this class will last more than 1.5 years?
The lifetime of a particular type of fluorescent lamp is exponentially distributed with expectation 1.6 years. [...]
Suppose that the lifetime of a particular brand of light bulbs is exponentially distributed with mean of 400 hours. [...]
Naturally, real lamps decay over time, so their lifetimes can't be memoryless. With that being said, is the exponential distribution a good approximation for the statistical lifetimes of any real lamps? Why/why not? If not, are there any other macroscopic systems whose lifetimes are modeled well by the exponential distribution?

etotheipi

## Answers and Replies

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Nugatory
Mentor
Many electronic and mechanical products (disk drives, for example) fail according to a "bathtub" curve. When first placed into service, the failure rate is high as units with manufacturing defects cull themselves. After that the failure rate is low and essentally flat until components reach their design lifetime and wear out; the failure rate then climbs sharply.

Vanadium 50, Lord Jestocost and berkeman
Have a look at: H.S. Leff, 1990. Illuminating physics with light bulbs. The Physics Teacher, 28, 30-35
Illuminating physics with light bulbs

The exponential distribution isn't a good approximation for the statistical lifetimes of real light bulbs. In case the failure rate increases with time due to aging, on uses the so-called "Weibull" family of distributions: https://www.mathpages.com/home/kmath122/kmath122.htm
Fascinating, thank you! I would have expected to see something more like the bathtub curve that Nugatory mentioned. Interesting derivation of the Weibull distribution's CDF F(𝜏) in the Illuminating physics article (well, 1-F(𝜏), named the surviving fraction).

At least the exponential distribution kind of makes sense in that it's the Weibull distribution with shape parameter k=1, whereas the case of k=4 seems to be much more relevant for real incandescent light bulbs. Since the Weibull PDF is so different for these two choices of k, it seems that authors of problems like the ones in my original post should at least mention it in the footnote.