I’ve been taking a grad course in statistics this semester and so I’ve been thinking about all sorts of real world examples of math, including the classic product-failure example that’s a mainstay of most stat classes.
One of the simplest continuous distributions is the exponential distribution which is a pretty decent way to model product failures. The probability of failure after time is given by
I read this great article about product failure and testing in Wired this week. I encourage you to check it out. Read the last page of the article especially, where it talks about how cutting-edge companies are modeling minute variations in materials using an electron microscope and some statistics. Instead of actually testing the product over and over again using a fatigue machine, they can create surprisingly accurate models of the materials using computers. Prior to this, the behavior of materials was somewhat unpredictable.
Of course I was excited to see this figure in the article, which shows the Weibull distribution modeling failures of steel bars in a fatigue machine.
The Weibull distribution, unlike the exponential distribution, takes the age of a product into account. If the parameter is greater than zero, than the rate of product failure increases with time. The probability of failure after time is
The first obvious thing to note is that the exponential distribution is just a special case of the Weibull distribution, with . The next thing to say is that this distribution is single-peaked. So how is the above a Weibull distribution? The article says it is, but I think it might be a linear combination of two Weibull distributions, don’t you? Whatever–normalize, and you’ve got yourself a probability distribution.
The real question is, if this is TWO Weibulls, would you settle for the lesser of two Weibulls?
Sorry. I had to.