In addition to eating too much, watching too much rubbish tv, and maybe drinking a little too much too, I have been reading a paper on the statistics of times-between-failures of air conditioning units of Boeing 720 airliners. These airliners are now obsolete, as I guess are the air conditioning units – the paper is from 1963. But the paper is a classic, the maths in it is relatively formal but the idea is simple.
The author of the paper, Proschan, worried about whether or not it was OK to lump together failure times from different aircraft together, and so to implicitly assume that all the air conditioning units in all the aircraft are the same. This sounds quite trivial but the failures have a large random element, and this makes it actually quite hard to know whether there are real differences from one plane to another or whether one plane has more failures just by chance. Proschan looked at how you would quantitatively say whether or not the members of randomly failing objects are really the same or whether any differences between their failures are just due to chance.
This need to tell if things are really different or not is actually very common. For example, say you are peeling sprouts from Waitrose while your friend is peeling sprouts from Aldi. You both notice that some of the sprouts you are peeling are off, and wonder whose sprouts are worse for this, Aldi’s or Waitrose’s? You both count how many sprouts are good between each off sprout. For example, maybe you peel 8 good ones, then an off one, then 11 good ones, another off one, etc. Proschan was one of the first to show a simple clear way how to use these numbers to find out, in this case, whether Aldi and Waitrose sprouts are really different or not.
Although I actually like sprouts, I don’t want to apply Proschan’s idea to sprouts. I want to apply them to the formation of crystals.
I am interested in the nucleation of crystals in sets of small droplets. These droplets look the same, but are they? I think in many cases they are not because, just by chance one droplet may have a piece of dirt in it too small to be seen, that is still big enough to speed up nucleation in that droplet. I think Proschan’s key result may be handy here. Basically what he did is show that if the distribution of times in individual objects (A/C units in his case, droplets in mine) is exponential, then if the units are different when you lump the times together the resulting distribution has to curve up with respect to an exponential. Again this may sound boring but in practice this is very handy, all you have to do is plot the distribution, compare it to an exponential and you have what you need to know.