 # Maths at Bondi Beach  ## Probability Theory

This is a work in progress. I am first publishing some minor results in relation to Gaussian theory (at university level) but the real aim is to comprehensively go through combinatorial theory for a start and then slide into probability theory. I will use Kai Lai Chung's proof of the probability that the sun will rise tomorrow in the context of demonstrating some combinatorial techniques.

The following downloads are university level and relate to some basic results.

1.Completing the square in the context of Gaussian integrals.  This is really fundamental even though it is not profound. I have included a standard result for bivariate normal distributions. Download

2.Demonstrating rigorously  how a characteristic function of a Gaussian density is derived using complex variable theory. You do this once in your life but having done it you know why it works. Download

3. Exponential waiting times: this is a quick derivation of  the distribution by taking a discrete Bernoulli process and then using an approximation style of argument.  It is a standard sort of thing one should know. Download

4. Sums of normal variables: This short paper uses three problems from William Feller's probability textbook to arrive at an approximation originally derived by Lagrange. The paper refers to a more detailed paper by George Marsaglia dealing with sums  and ratios of sums of normal variates. Probability generating functions are employed.  Download

5. The fallacy of time diversification – what financial planners don’t understand.  Download

6. How long will my savings last?  A simple application of the annuity formula.  Download

7. What do the arc sine law and schmucks have in common? In this article I explore the counter-intuitive nature of the arc sine law in run theory and its implications for investment.  Download

8.  I have uploaded a detailed paper on Laplace's Law of Succession ("What is the probability that the sun will rise tomorrow?"). This paper is based on Kai Lai Chung's rederivation of the relevant probability by using an urn model.  I have gone through all the intricate combinatorial steps so if you want to see all the gory details read on. A much simpler continuous derivation is also given. Laplace's Law of Succession.pdf

9. Chebyshev’s inequality with a hint of measure theory.  The Rigour Police will come around and beat me senseless for this but here goes anyway. Who knows, someone may find it useful.  Chebyshev

10. In his famous probability textbook, William Feller derived a surprising result in relation to the variance of n mutually independent random Bernoulli variables with variable probabilities of success. If we focus on Bernoulli trials with a constant probability of success which equals the average of the variable probabilities we get the counterintuitive result that the variability of the original probabilities, or lack of uniformity, decreases the magnitude of chance fluctuations as measured by the variance,  Thus given a certain average quality p of n machines, the output will be least uniform if all machines are equal. Think of how this principle applies to investment managers and MOOCs (massive on-line open courses).  To read the detailed argument and derivations click here:

Bernoulli trials with variable probabilities - an observation by Feller.pdf

11.Australian mathematician/statistician Peter Donnelly who works at Cambridge University demonstrates in a TED talk (http://www.ted.com/talks/peter_donnelly_shows_how_stats_fool_juries?language=en#   )  just how poorly the judicial system handles probabilistic and statistical arguments.  He does this by posing a simple coin tossing experiment and then develops how the English  court system put a mother in gaol for the alleged murder of her two children because of some faulty statistical arguments.  To understand how juries  and judges can be fooled by statistics click Fooling juries with statistics.pdf

12. If you have ever wondered how Laplace actually proved the Central Limit Theorem you can see how it was done  by downloading my explanation of how he did it: The central limit theorem - how Laplace actually proved it.pdf

13. Rademacher functions are simple step-functions which have a surprising role in the concept of statistical independence.  This role was explored by Mark Kac in his well known book “Statistical Independence in Probability, Analysis and Number Theory”.  One of the basic properties of Rademacher functions is their orthogonality (actually orthonormality) and in this paper I demonstrate a purely combinatorial proof of the orthogonality and contrast it with the traditional proofs.  To view the paper click: A combinatorial view of the orthogonality of Rademacher functions.pdf

14. For a detailed exposition of Vieta's formula with its implications for probabilistic approaches to number theory see the attached paper which is based on Mark Kac's famous book on number theory and statistical independence:  Vieta's formula

15.  