[[!meta title="Gumbel/Fisher-Tippett distributions"]] [[!meta date="2008-06-29 18:10:54"]] Aha, the probability of Bell-model unfolding under constant force-loading has a name! It is a [Fisher-Tippett distribution](http://en.wikipedia.org/wiki/Fisher-Tippett_distribution), of which the Gumbel distribution is a particular type. However, NIST refers to it as a [minimum Gumbel distribution](http://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm). Hmm, hopefully I'm not just confusing myself looking at the standardized form, let me go double check... What is a [cumulative distribution function]() anyway? Ah, `CDF(x)` is just the probability that the variable will be `<= x`, so the probability distribution function is given by `PDF(x) = -d(CDF)/dx`. Alright, looks like my distribution is a bit different than the Fisher-Tippett because I need a non-unity a factor `a` in `PDF(x) = exp(-ax/b)*exp[-exp(x/b)]` with `z := exp(-x/b)`. Basically, I have a Fisher-Tippett distribution with a poorly scaled `x`, but I don't know how to rescale `x` until I've fit my distribution. So the search continues... The [Gompertz-Makeham Law](http://en.wikipedia.org/wiki/Gompertz-Makeham_law_of_mortality) law for exponentially increasing failure rate is what I want. But Wikipedia says this is the same as Fisher-Tippett with time inversion. There is a nice discussion of aging in general, but not much math [here](http://longevity-science.org/Failure-Models-2006.pdf). Update: Related posts: * [[Gompertz-Gumbel distributions|Gompertz-Gumbel_distributions]] * [[Gompertz paper'|Gompertz_paper]] * [[Giving up on Gompertz theory]] [[!tag tags/theory]] [[!tag tags/sawsim]]