Binomial trees and a closed form expression for the terms of a recurrence relation

Recently, while playing with an integral of a probabilistic function possibly inspired by the binomial trees I’ve been reading about in financial mathematics books, I was attempting to find a closed form expression for the following recurrence relation: 

Let c_k(n), k, n = 0, 1, 2, …, satisfy

c_k(n) = c_k-1(n) + c_k-1(n-k)

subject to the boundary conditions

c₀(0) = 1 and c₀(n) = 0 for all n=1, 2, …

Fun problem to tackle

Here’s a little maths problem to tackle:

Occupied by other interests

It’s been ages since I wrote my last post – other interests have crowded out my blogging time. It’s been interesting not blogging much – I’m enjoying other activities.

I am now at my workplace Mon-Thur each week – Friday is my day for doing scientific/mathematical or other activities – who knows what I may do? I might end up running my own business! I think that many people would like to work without being an employee. I understand its attraction to people who are employees.

Feel free to leave any business proposals in comments.

Using funds in superannuation accounts for first-property deposits?

I had an idea for a potential policy and I’ll write it here to keep a record of it.

Using superannuation for first-property deposits. While the purpose of super is savings for old age (and reduced dependence on the state), people might find it difficult to enter the property market in Australian capital cities due to high house prices. Compulsory super has resulted in many workers having a pool of super.

Strong reciprocity from “The Origin of Wealth”, by Eric D. Beinhocker

I’m reading a number of books on economics & finance at the moment including The Origin of Wealth by Eric D. Beinhocker, the subtitle of which Evolution, Complexity and the radical remaking of Economics describes its focus.

In this book, Beinhocker describes developments in the field of complexity economics in the last 15-odd years, which, in my understanding, seems to be about conceptualising economics arising from interacting agents each possessing incomplete information and processing capabilities.

The novel is very interesting - here is an excerpt on the fundamentals of political ideologies (the Left/Right divide) from pages 418-421.

Expectation value of a coin flipping game (the tools of a mathematician)

was recently asked how I might go about working out the expectation value of the game where an unbiased coin is thrown n times where you win nothing each time a head is thrown and you win $n if you throw a tail on the n^th throw. 

I hadn’t thought of this problem for a long time and didn’t have a pen and paper on hand and couldn’t give an answer (!) - but when I got home, quickly playing with the problem with pen and paper showed that, of course, the expectation value is
$n(n + 1)/4. 

Pen and paper – the tools of a mathematician! It would be nice to have a mental whiteboard on which to do calculations, but paper is much more reliable!

Piers Akerman’s piece on the Medically Supervised Injecting Centre

Piers Akerman has written a piece on his Daily Telegraph blog on the Kings Cross Medically Supervised Injecting Centre at this link.

I twice tried to leave a comment on his blog, but both times received the error message “You are not authorised to do that action” when I pressed the submit button” (I wonder what you have to do to be “authorised”?!) 

Here is the comment I submitted to the blog:

Piers, as someone who has submitted mathematics papers to journals for publication, I know the importance of peers reviewing work. Non-experts don’t know if the authors of the review used a sound methodology, nor if they did a proper analysis. I am in favour of evidence-based public policy, and proper reviews of any public policy are extremely valuable. The “review” mentioned in this piece may be valuable, but non-experts have no way of judging this (eg judging its methodology).

Given that the conclusions of this review are at odds with many people think, it is important that people who are able to judge its methodology do so before the review is used as a basis of public policy.

Terence Tao’s blog

I’ve come across Terence Tao’s blog and put a link to it on my blogroll. Terry is of course one of the 2006 Fields Medallists. Enjoy reading his blog.