Financial Mathematics: Statistics - Properties of Distributions

Probability distributions have a number of properties which help us summarize and characterize them. We'll look at some of these properties, how they are calculated and what they are used for.

Contents:

I) Mean
II) Median
III) Variance
IV) Skewness
V) Kurtosis

On Counting (Exploring Operational Definitions Part III)

For the first two parts in this series see:

Exploring Operational Definitions: Part I
Exploring Operational Definitions: Part II - Distance

Perhaps the "simplest" procedure that most folks have learned is the technique(s) of counting. What I would like to explore is that there are a variety of techniques that we call counting. In some cases they build on one another. In other cases, they are techniques which give "approximate" solutions.

Of course not all societies count things (see here). Nonetheless, I suspect that many of our "intuitions" about mathematics ultimately stem from our earlier experience with counting. Our attachment to such intuitions will somewhat determine how willing we are able to accept alternative definitions and techniques for counting. Today I'll explore a few of these definitions.

A Unique Solution to the Monty Hall Problem

The Monty Hall Problem is a counterintuitive result in statistics. The problem goes something like this.

Sketch of the Monty Hall Problem

Suppose there are three doors: A, B and C. Behind one of the doors is a big prize (a car, a huge pile of cash). Behind the other two doors are just some goats.

Now you make a selection of a door but the game is not over with yet. Suppose you pick door letter A. Now the game is not over with yet at this time. The host actually reveals one of the remaining doors, say door letter C, to have a goat.

Now the host gives you the option: stick with door A or switch to door B.

On [Logical] Equivalence

In Newton's Law is F=ma?, explored the issue of whether or not this formulation was equivalent to Newton's actual statement of his 2nd Law. Today I want to further explore the topic of [logical?] equivalence. In general, I want to know what it means to say that two statements, $P$ and $Q$, are equivalent.

Growth at 2% Per Year

So one thing that frequently occurs is that we assume growth will continue at a constant rate. I mention 2% because this is often seen as the long-term real growth rate of an economy. What I want to do is get some perspective on that.

Let's suppose we're living 6000 years ago, circa the 4th century BCE. If I started with 1 of something (it doesn't matter at this point what it is) 6000 years ago, and it grew at 2% per year, how much would I have today? The answer is simple:

Poincare on Two Types of Mathematicians

In Henri Poincaré's book, Science and Method, there is an interesting discussion on mathematics education. I'll quote a relevant portion:

Many children are incapable of becoming mathematicians who must none the less be taught mathematics; and mathematicians themselves are not all cast in the same mould. We have only to read their works to distinguish among them two kinds of minds - logicians like Weierstrass, for instance, and intuitionists like Riemann. There is the same difference among our students. Some prefer to treat their problems "by analysis," as they say, others "by geometry". 

It is quite useless to seek to change anything in this, and besides, it would not be desirable. It is well that there should be logicians and that there should be intuitionists. Who would venture to say whether he would prefer that Weierstrass had never written or that there had never been a Riemann? And so we must resign ourselves to the diversity of minds, or rather we must be glad of it.

Grassmann's Real versus Formal distinction

So I'm currently reading A History of Vector Analysis written by Michael J. Crowe (or to say it another way, I'm not normal). I came across the book after reading this article: Hermann Grassmann and the Creation of Linear Algebra which I read because I have an interest in the development of linear algebra (aka I'm not normal).

Crowe quotes at some length from a book written by Grassmann entitled: Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durchAnwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik,Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert (which in German means tl;dr). But he makes the following distinction:

The primary division in all the sciences is into the real and the formal. The former represent in thought the existent as existing independently of thought, and their truth consists in their correspondence with the existent. The formal sciences on the other hand have as their object what has been produced by thought alone, and their truth consists in the correspondence between the thought processes themselves.

From what I'm gathering from what little bits Crowe has quoted, Grassmann thinks that his system (which from what I gather was an early form of modern day linear algebra) as being merely "formal". It's existence has been posited through pure thought but it doesn't correspond to anything in the world (outside of the correspondence to human thought). Contrast that with geometry which corresponds with space and is therefore an example of a "real" science. He even went so far as to suggest that geometry properly ought not be included within mathematics:

It had for a long time been evident to me that geometry can in no way be viewed, like arithmetic or combination theory, as a branch of mathematics; instead geometry relates to something already given in nature, namely, space.

What he calls "pure mathematics", by contrast is:

... the science of the particular existent which has come to be through thought. The particular existent, viewed in this sense, we name a thought-form or simply a form. Thus pure mathematics is the theory of forms.

I'm not sure how far these distinctions would go today. Consider space for example. In a post-Einstein world, space has different structure when in the presence of objects of mass. Mass actually bends or warps space. And it's not entirely restricted to 3 dimensions as it was in Grassman's day.

On the more "abstract" side of things, the lines are even more blurred. We now deal with infinitely dimensional complex vector spaces to explain such phenomenon as matter waves.

At this point there should be a cue for a joke about Hilbert spaces but I'm too lazy to write one. So I'll just get a room in his hotel and crash for the night.

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Was Chess Invented or Discovered?

One common comparison made within philosophy of math circles is a comparison between the game of chess and mathematics. Variants of, in particular, formalism hold that mathematics is a sort of game with well defined rules that mathematicians play. I think the game of chess is actually quite apt for the comparison. Like all metaphors, it has limitations but I think it's still an interesting one.

The closest comparison, in my view, is between chess and geometry. Here's how it works.

Being Without Numbers

Linguistic anthropologist Daniel Everett has been researching and living with (off and on) an Amazonian tribe called the Pirahã for a few decades now. In Don't Sleep, There Are Snakes: Life and Language in the Amazonian Jungle, Everett gives a semi-autobiographical account of his interactions with the Pirahã.