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.

Recall that the Baron’s game consisted of guessing under which of a pair of cups was to be found a token for a stake of four cents and a prize, if correct, of one. Upon success, Sir R----- could have elected to play again with three cups for the same stake and double the prize. Success at this and subsequent rounds gave him the opportunity to play another round for the same stake again with one more cup than the previous round and a prize equal to that of the previous round multiplied by its number of cups.

A few years ago we saw how we could approximate a function f between pairs of points (x_i, f(x_i)) and (x_i+1, f(x_i+1)) by linear and cubic spline interpolation which connect them with straight lines and cubic polynomials respectively, the latter of which yield smooth curves at the cost of somewhat arbitrary choices about their exact shapes.
An alternative approach is to construct a single function that passes through all of the points and, given that n^th order polynomials are uniquely defined by n+1 values at distinct x_i, it's tempting to use them.

Recently my fellow students and I have been spending our free time using Professor B------'s remarkable calculating engine to experiment with cellular automata, being mathematical contrivances that might be thought of as crude models of the lives of those most humble of creatures; amoebas. In their simplest form they are unending lines of boxes, some of which contain a living cell that at each generation will live, die or reproduce according to the contents of its neighbouring boxes. For example, we might say that each cell divides and its two offspring migrate to the left and right, dying if they encounter another cell's progeny.