This time we shall see how we can approximate the function that defines the relationship between them without actually revealing what it is. ]]>

Most recently we have seen how we can categorise automata by the manner in which their populations evolve from a primordial state of each box having equal chances of containing or not containing a cell, be they uniform, constant, cyclical, migratory, random or strange. It is the latter of these, which contain arrangements of cells that interact with each other in complicated fashions, that has lately consumed our attention and I shall now report upon our findings. ]]>

Will you also accept a wager to warm your blood?

It gladdens my heart to hear so sir!

I propose a game that oft puts me in mind of the banquet held in the great hall upon Mount Olympus to which I was invited as the guest of honour by Zeus himself! ]]>

This time we shall take a look at the binomial distribution which governs the number of successes out of

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. ]]>