My fellow students and I have lately been spending our spare time experimenting with cellular automata, which are simple mathematical models of single celled creatures such as amoebas, governing their survival and reproduction from one generation to the next according to the population of their neighbourhoods. In particular, we have been considering an infinite line of boxes, some of which contain living cells, together with rules that specify whether or not a box will be populated in the next generation according to its, its left hand neighbour's and its right hand neighbour's contents in the current generation.
We have found that for many such automata we can figure the contents of the boxes in any generation that evolved from a single cell directly, in a few cases from the oddness or evenness of elements in the rows of Pascal's triangle and the related trinomial triangle, and in several others from the digits in terms of sequences of binary fractions.
We have since turned our attention to the evolution of generations from multiple cells rather then one; specifically, from an initial generation in which each box has an even chance of containing a cell or not.

Last time we took a first look at Bernoulli processes which are formed from a sequence of independent experiments, known as Bernoulli trials, each of which is governed by the Bernoulli distribution with a probability p of success. Since the outcome of one trial has no effect upon the next, such processes are memoryless meaning that the number of trials that we need to perform before getting a success is independent of how many we have already performed whilst waiting for one.
We have already seen that if waiting times for memoryless events with fixed average arrival rates are continuous then they must be exponentially distributed and in this post we shall be looking at the discrete analogue.

Sir R----- my fine friend! Will you take a glass of perry with me to cool yourself from this summer heat?

Good man!

Might I also presume that you are in the mood for a wager?

Stout fellow!

I suggest a game that ever puts me in mind of that time in my youth when I squired for the warrior king Balthazar during his pilgrimage with kings Melchior and Caspar to the little town of Bethlehem.

Several years ago we took a look at memoryless processes in which the probability that we should wait for any given length of time for an event to occur is independent of how long we have already been waiting. We found that this implied that the waiting time must be exponentially distributed, that the waiting time for several events must be gamma distributed and that the number of events occuring in a unit of time must be Poisson distributed.
These govern continuous memoryless processes in which events can occur at any time but not those in which events can only occur at specified times, such as the roll of a die coming up six, known as Bernoulli processes. Observations of such processes are known as Bernoulli trials and their successes and failures are governed by the Bernoulli distribution, which we shall take a look at in this post.

In their most recent wager Sir R-----'s goal was to guess the outcome of the Baron's roll of four four sided dice at a cost of four coins and a prize, if successful, of forty four. On the face of it this seems a rather meagre prize since there are two hundred and fifty six possible outcomes of the Baron's throw. Crucially, however, the fact that the order of the matching dice was not a matter of consequence meant that Sir R-----'s chances were significantly improved.

Several years ago we took a look at the gamma function Γ, which is a generalisation of the factorial to non-integers, being equal to the factorial of a non-negative integer n when passed an argument of n+1 and smoothly interpolating between them. Like the normal cumulative distribution function Φ, it and its related functions are examples of special functions; so named because they frequently crop up in the solutions to interesting mathematical problems but can't be expressed as simple formulae, forcing us to resort to numerical approximation.
This time we shall take a look at another family of special functions derived from the beta function B.

You will no doubt recall my telling you of my fellow students' and my latest pastime of employing Professor B------'s Experimental Clockwork Mathematical Apparatus to explore the behaviours of cellular automata, which may be thought of as simplistic mathematical simulacra of animalcules such as amoebas.
Specifically, if we put together an infinite line of imaginary boxes, some of which are empty and some of which contain living cells, then we can define a set of rules to determine whether or not a box will contain a cell in the next generation depending upon its own, its left and its right neighbours contents in the current one.

In the previous post we explored the Cauchy distribution, which, having undefined means and standard deviations, is an example of a pathological distribution. We saw that this is because it has a relatively high probability of generating extremely large values which we concluded was a consequence of its standard random variable being equal to the ratio of two independent standard normally distributed random variables, so that the magnitudes of observations of it can be significantly increased by the not particularly unlikely event that observations of the denominator are close to zero.
Whilst we didn't originally derive the Cauchy distribution in this way, there are others, known as ratio distributions, that are explicitly constructed in this manner and in this post we shall take a look at one of them.

Sir R-----! Come join me for a glass of chilled wine! I have a notion that you're in the mood for a wager. What say you?

I knew it!

I have in mind a game of dice that reminds me of my time as the Russian military attaché to the city state of Coruscant and its territories during the traitorous popular uprising fomented by the blasphemous teachings of a fundamentalist religious sect known as the Jedi.

Last time we took a look at basis function interpolation with which we approximate functions from their values at given sets of arguments, known as nodes, using weighted sums of distinct functions, known as basis functions. We began by constructing approximations using polynomials before moving on to using bell shaped curves, such as the normal probability density function, centred at the nodes. The latter are particularly useful for approximating multi-dimensional functions, as we saw by using multivariate normal PDFs.
An easy way to create rotationally symmetric functions, known as radial basis functions, is to apply univariate functions that are symmetric about zero to the distance between the interpolation's argument and their associated nodes. PDFs are a rich source of such functions and, in fact, the second bell shaped curve that we considered is related to that of the Cauchy distribution, which has some rather interesting properties.