Welcome Sir R-----! Pray shed your overcoat and come dry yourself by the fire. I am told that these spring showers are of inestimable benefit to farming folk, but I fail to grasp why they can’t show the good manners to desist until noblemen have made their way indoors.

Will you join me in a warming measure and perchance a small wager?

I had no doubt sir!

I’ve a mind for a game oft played by the tribesmen of Borneo upon the cobbled floors of their homes as a means of practice for their legendary talent in forging paths through the dense forests in which they dwell.

A few years ago we spent some time implementing a number of the sorting, searching and set manipulation algorithms from the standard C++ library in JavaScript. Since the latter doesn't support the former's abstraction of container access via iterators we were compelled to restrict ourselves to using native Array objects following the conventions of its methods, such as slice and sort.
In this post we shall take a look at an algorithm for finding the centrally ranked element, or median, of an array, which is strongly related to the ak.nthElement function, and then at a particular use for it.

Recall that in the Baron's latest wager, Sir R-----'s goal was to traverse a three by three checkerboard in steps determined by casts of a four sided die, each at a cost of two coins. Moving from left to right upon the first rank and advancing to the second upon its third file, thereafter from right to left and advancing upon the first file and finally from left to right again, he should have prevailed for a prize of twenty five coins had he landed upon the top right place. Frustrating his progress, however, were the rules that landing upon a black square dropped him back down to the first rank and that overshooting the last file upon the last rank required that he should move in reverse by as many places with which he had done so.

Last time we saw how we can use the chi-squared distribution to test whether a sample of values is consistent with pre-supposed expectations. A couple of months ago we took a look at Student's t-distribution which we can use to test whether a set of observations of a normally distributed random variable are consistent with its having a given mean when its variance is unknown.

Recently, my fellow students and I have been caught up in the craze that is sweeping through the users of Professor B------'s clockwork calculating engine; namely the charting of sets of two dimensional points that have fractal planar boundaries, being those that in some sense have a fractional dimension. Of particular interest have been the results of repeated applications of quadratic functions to complex numbers; specifically in measuring how quickly, if at all, they escape a region surrounding the starting point, by which charts may be constructed that many of the collegiate consider so delightful as to constitute art painted by mathematics itself!

A few months ago we explored the chi-squared distribution which describes the properties of sums of squares of standard normally distributed random variables, being those that have means of zero and standard deviations of one.
Whilst I'm very much of the opinion that statistical distributions are worth describing in their own right, the chi-squared distribution plays a pivotal role in testing whether or not the categories into which a set of observations of some variable quantity fall are consistent with assumptions about the expected numbers in each category, which we shall take a look at in this post.

Greetings Sir R-----! Come warm yourself by the hearth and take a dram of scotch!

Would you care for a wager to fire up your blood?

Stout fellow!

I propose a game that puts me in mind of an ill-fated caving expedition that I undertook some several years ago.

Last time we took a look at the chi-squared distribution which describes the behaviour of sums of squares of standard normally distributed random variables, having means of zero and standard deviations of one.
Tangentially related is Student's t-distribution which governs the deviation of means of sets of independent observations of a normally distributed random variable from its known true mean, which we shall examine in this post.

Most recently the Baron challenged Sir R----- to a race of knights around the perimeter of a chessboard, with the Baron starting upon the lower right hand square and Sir R----- upon the lower left. The chase proceeded anticlockwise with the Baron moving four squares at each turn and Sir R----- by the roll of a die. Costing Sir R----- one cent to play, his goal was to catch or overtake the Baron before he reached the first rank for which he would receive a prize of forty one cents for each square that the Baron still had to traverse before reaching it.

Several years ago we saw that, under some relatively easily met assumptions, the averages of independent observations of a random variable tend toward the normal distribution. Derived from that is the chi-squared distribution which describes the behaviour of sums of squares of independent standard normal random variables, having means of zero and standard deviations of one.
In this post we shall see how it is related to the gamma distribution and implement its various functions in terms of those of the latter.