This time we shall take a look at the distribution of the number of failures before a given number of successes, which is a discrete version of the gamma distribution which defines the probabilities of how long we must wait for multiple exponentially distributed events to occur. ]]>

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. ]]>

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. ]]>

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. ]]>

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. ]]>

This time we shall take a look at another family of special functions derived from the beta function B. ]]>

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. ]]>

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. ]]>

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. ]]>

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. ]]>

An alternative approach is to construct a single function that passes through all of the points and, given that