Aha, the probability of Bell-model unfolding under constant force-loading has a name! It is a Fisher-Tippett distribution, of which the Gumbel distribution is a particular type. However, NIST refers to it as a minimum Gumbel distribution.

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 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.

Update: Related posts: