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I'm a student wanting to write a report in something that has to do with mathematical knowledge. I came on this website on saw an answer by Stefan Hansen, which I would like to implement in my paper.

Counting processes and martingales are objects I view as purely mathematical/probabilistic objects. Nevertheless, they fundamental objects when describing the theory of survival analysis - survival analysis being a branch that is used in many registry-based studies in e.g. epidemiology.

A simple model (a model without censoring) of survival analysis is the following: Let X1,…,Xr be iid random variables with values in (0,∞), where Xi is the lifetime of the ith individual. Let Xi have density f and distribution function F with F(t)<1 for all t∈(0,∞). Put Nit=1{Xi≤t},i=1,…,r, and Nt=∑i=1rNit, i.e. Nt is the number of individuals dead before t. Then (N1t,…,Nrt)t≥0 is an r-dimensional counting process and (Nt)t≥0 is a counting process. Now, theory of local martingales and predictable covariation can be used to derive estimators such as the Nelson-Aalen estimator of the cumulative hazard rate, i.e. the function Λ(t)=−logS(t), where S(t)=1−F(t) is the survival function.

I was wondering if one could explain this at a middle school level. And it would be great if I could get into contact with Mr. Hansen.

After talking with @SimonS, I decided to reinstate my question. I want to show some easy pure mathematics that is not at first sight applied, but can be applied (survival analysis, for example). I want to actually show the maths, but this needs to be readable at the high school level. Thanks again.

Thanks,

hariq
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  • For middle school you are overshooting. Martingales are at a level of sophistication beyond high school level. I suggest you find another topic which you already know something about and ask again if you have a specific question about that new topic. – Simon S Sep 23 '15 at 13:20
  • @SimonS Thanks Simon, I appreciate your kind input here. My paper isn't really about mathematics, but about how how pure mathematics can be applied in the long run of things. So, its more a philosophy question and paper. I wanted to use an example like the one above, and then demonstrate it to show the applicability of pure mathematics. I know that there are many examples, but most above my level. Could you kindly suggest anything? – hariq Sep 23 '15 at 14:43
  • If you search for "number theory and cryptography" you will find a lot of material. Number theory was famously developed well beyond application--considered then and now an exemplar of pure mathematics. Today number theory it is the backbone of nearly all communication encryption. The basics of number theory are also quite accessible. – Simon S Sep 23 '15 at 14:46
  • @SimonS Thanks Simon. This helps out a bit and I'll check it out. – hariq Sep 23 '15 at 15:10
  • In the middle school I attended, students would not know what $S(t)$ means, or what $i=1,...,r$ means, or what $\log$ is, or what $(0,\infty)$ means, or what $\Sigma$ means, or what $X_i$ (subscript notation to get new variables) means, etc. Also, in today's PC climate (and maybe even in my early 1970s middle school climate), you definitely wouldn't want to say anything like "number of individuals dead before $t$". – Dave L. Renfro Sep 24 '15 at 19:46

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There is a wonderful example, of Penrose's aperiodic pentagonal tilings. It seems completely irrelevant to anything - but Kimberley-Clarke, the toilet paper manufacturer, used it as the print on their paper so that when it was all rolled up for selling, there was no possibility that successive layers would get stuck together. There would always be a pattern of ridges in the paper, keeping the layers slightly apart.

Patrick Stevens
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  • Wow! This is a great example. Thanks Stevens. Could you please explain this a bit deeper and in layman’s terms for me? Other than that this is pretty interesting and I appreciate your answer. – hariq Sep 25 '15 at 01:52
  • I don't know it any deeper than the Wikipedia page goes into, and that's pretty much already in terms as layman as it gets. – Patrick Stevens Sep 25 '15 at 08:09