From the Exploring Data website - http://curriculum.qed.qld.gov.au/kla/eda/
© Education Queensland, 1997

Light Bulbs and Dead Batteries

Don Kerr, of Brisbane-based Zeno Educational Consultants, once told me that he believed that the lifetime of light bulbs and car batteries both have a decaying exponential distribution. I was intrigued by this, as every statistics textbook I have ever used always had a question that started, ‘Assume the lifetime of light bulbs is normally distributed, with mean....’.

So I decided to ask my colleagues on the EdStat mailing list.

The Original Email

In all of my old stats textbooks, in the chapter on the normal distribution, there were always some questions on the lifetime of light bulbs, usually containing the phrase, 'Assume that the lifetime of light bulbs is normally distributed...'

As these books were written (I assumed) by statisticians who would obviously have studied such things, I never questioned this assumption - and it seemed reasonable enough that somehow the filament gradually wore out and that the distribution of lifetimes of bulbs was hence normally distributed.

But according to some recent information I have received, the distribution may be better modelled by a decaying exponential function - not normal at all. In fact Gordon Smyth at University of Queensland says that the actual distribution is only roughly exponential - tests done by Choice magazine (similar to Consumer Reports in the US) show that under 1000 hours, fewer bulbs fail than would be expected, but over 1000 hours the bulbs fail more rapidly than expected.

Now I find this very interesting because statistically a decaying exponential function to me implies a random process rather than a gradual wearing out of a component. For example, the half-life of a radioactive substance and the distribution of waiting times are decaying exponential functions and are governed by random processes. So the implication is that the lifetime of a light bulb is similarly governed by random processes. It also implies that some bulbs would have an incredibly long lifetime.

I have also heard the claim that the lifetime of car batteries has the same distribution - decaying exponential. I would have assumed that batteries just wore out and a normal distribution (or maybe a slightly skewed distribution) would have been an appropriate model.

There is an interesting repercussion if the life of car batteries is decaying exponential. Say the average life of a car battery is 3 years, and someone steals your car's battery after 2.5 years. Your insurance policy states that you will be reimbursed for the 'market value' of the stolen item. The insurance company claims that the battery has on average only half a year of life remaining and hence offers to pay for one-sixth of the cost of a replacement battery. You claim that since the distribution of battery lifetimes follows a decaying exponential function, the average lifetime of the battery measured from the time of the theft is still 3 years - this is a property of decaying exponential functions. Therefore you should receive a brand new battery. Interesting, eh? Some years ago something like this actually happened to a colleague, who wrote a letter to an insurance company countersigned by the Head of Mathematics at UQ. The insurance company wrote back, 'We have no idea what you are talking about. We're not paying.'

Does anyone have any knowledge on these matters?

Some Responses

From Bob Hayden

Rex, burn those books!

From ?

Rex: you may in fact be right on about the decaying exponential function, but I think you have a snowball's chance in hell of getting insurance companies to buy you a new battery. :-)

From Tim Erickson

I love it!

I have no REAL knowledge of these matters, but let me put another in your hopper:

When you buy a hard disk drive, they give you a figure -- often called, pompously, a mean-time-between-failures (MTBF) -- of a large number of hours (80,000, say). You say, "Wow! I can leave this drive on for ten years!" The implication is that, of course, they'll last about ten years, plus or minus something, because they'd be normally distributed, blah blah blah.

Isn't it upsetting when it fails in one year plus epsilon (right after the warranty runs out)? Did you get the five-sigma lemon?

But wait: they only just designed this drive. How do they know it'll last ten years? My informant told me that they take a thousand drives (say) and run them for a month. Nine fail. They have 720,000 drive-hours, and nine failures, so that's about 80,000 hours per failure.

That is, their test does not distinguish normal from exponential behaviour. Now, our everyday experience with friction and entropy suggests that, in a device with moving parts, they will generally wear OUT, not IN. So the rate of failure ought to increase with time, as Rex's original letter suggests.

I also wonder whether, since new things may have manufacturing flaws, the folks who determine MTBF run those thousand drives for a couple weeks first and exclude the ones that fail right away before they start their test.

Does anybody know how they really do these things? The quality-assurance people must have some model in order to budget for warranty repairs. And it seems to me the epistemology of manufacturers' claims might be an interesting topic for our students.

Ross A. Frick

The correspondence about the lifetimes of light bulbs is entertaining. It seems that nobody does any probability modelling any more! The modelling of lifetimes of various electrical or electromechanical devices is an interesting science in its own right, and ought to be a part of any undergraduate course in statistics, particularly for management and engineering degrees. I selected a book more or less at random (I mean, haphazardly!) from my bookshelf and found an excellent chapter titled "Reliability", in which are laid out the mathematics of the hazard function, the principles of lifetime testing and estimation of parameters in the Weibull distribution (a generalisation of the negative exponential). The book is by Lee J Bain and Max Engelhardt and titled "Introduction to Probability and Mathematical Statistics" (PWS-Kent). More accessible accounts can be found in many other texts written for undergraduate engineers and technologists, including a brief chapter in Christopher Chatfield "Statistics for Technology" (Chapman and Hall).

Back to light bulbs. The lifetime distribution is unlikely to be well represented by the negative exponential distribution. It is unreasonable to get a student to find out, because the lifetimes are generally large. However, I have had a project student find the best Weibull fit to the lifetimes of resistors under overload currents. The fit was good, and the distribution was not negative exponential. As long as the distribution is not negative exponential, all the nice ‘memoryless’ results are inapplicable, of course. [i.e., 'memoryless' in the sense that if the lifetime of light bulbs was truly negative exponential, then a bulb that had been burning for one thousand hours, say, would still have the same expected number of hours of use remaining as it had when it was first taken from its box.]

I am on the side of the insurance company in the dispute over the battery.

Summary

The logical next step in the investigation is to get some data on the distribution of the lifetimes of light bulbs, and car batteries, and determine for each how well a negative exponential function fits the data. I think an assignment based on the analysis of some real data in this area could make for interesting reading.