mathematics | Arbias Hashani

Why “Mean” And “Variance” Are So Important

July 6, 2013

Suppose you are at a football (soccer) match $X$ between teams $A$ and $B$ . You are interested in a gamble – it is assumed you want to earn money in the easiest way whilst adhering to the law), so you are offered a chance to gamble on the situation $X$ .

It is known that team $A$ is better than team $B$ with the additional information that team $A$ is very volatile and team $B$ is very consistent: $B$ stick to the same level most of the time where as $A$ can bounce up and down.

Which team would you gamble on to win the match?

What do you expect in return for your gamble?

Both choices poise a risk – $A$ is better so you should pick $A$ , where as $B$ is more consistent and $A$ is volatile, so you should pick $B$ .

Given an action, the possible outcome being undesirable is precisely the risk of the situation.

What we expect in return for our investment (say $R$ ) is the expected return $\mathbb{E}[R]$ . This is known as the mean of situation $X$ .

It is important as this value is what we get in return – a mean that is not desirable implies no gamble hence no investment. The mean is used as a measure of seeing success in an investment in this case.

Assume we have expected returns for team $A$ as $\mathbb{E}[R_A]$ and for team $B$ as $\mathbb{E}[R_B]$ .

We know that team $B$ is far more likely to achieve the expected return $\mathbb{E}[R_B]$ . As $A$ is more volatile, it is hard to trust (or perhaps even accept) $\mathbb{E}[R_A]$ .

So how do we determine how likely each team is to matching their expected return?

By looking at the variance $\mathbb{E}[(R-\mathbb{E}[R])^2]$ of $X$ . This looks at how far we expect to be away from the expectation as a form of a least square.

Then when we have the variance of each team, clearly the team with the lower variance is more desirable as they are more likely to achieve their expected return, which we would like to be as high as possible.

We translate these problems into mathematics by use of probability theory, to allow the chances of each situation happening and defining the specific functions for expected returns and variance.

We use statistics to find estimates of what we think the expected return and variance are, as they are usually unknown.

We use utility theory to understand the behaviour an investor should exhibit when going through these choices – if he sees two teams who give the same return, so supposing that they are equally as good as each other, then assuming no further information (such as one team being more consistent than the other), he should beat on either team without any worry, as they give the same return.

The behaviour should be consistent. The behaviour is explained by economics and is translated into mathematics by probability theory and statistics.

This is the kind of what actuaries do. Everything explained above can be seen as actuarial science. It is a simple example, but it shows why mean and variance are important and the motivations behind actuarial science.

Leave a Comment » | Mathematics | Tagged: actuarial, actuarial science, actuary, economics, mathematics, mean, Probability Theory, utility theory, variance | Permalink
Posted by AH

Transformers 3 Puzzle

January 3, 2013

The movie Transformers: Dark of the Moon contains an interesting puzzle.

Sam, the main character, has an evil robot (a small Decepticon) which transforms to a watch placed on his hand by Dylan, an evil guy.

The robot/watch can understand everything Sam says and hears. It reports information back to Dylan. It can tap into Sam’s nervous system and affect him, meaning he cannot just remove it. Sam must communicate to fellow humans and good robots (Autobot’s) but he knows the whole time whatever he hears or says is accessed to the enemy.

We can build an interesting question from this.

Suppose you know person X and Y, person X works for a competing, evil company X’ and person Y works for your employing, good company Y’. Suppose everything that you communicate with person Y goes to person X and you cannot inform person Y of this.

Devise a winning strategy to defeat company X’ and for company Y’ to win.

Related mathematical fields; algebra, probability theory, set theory, number theory.

Leave a Comment » | Mathematics | Tagged: Algebra, mathematics, number theory, operational research, Probability Theory, set theory, transformers | Permalink
Posted by AH

What is a good mathematical question?

December 27, 2012

What does a mathematician do? A good question.

One could say a mathematician observes and provides a logical input.

Another could say that mathematicians prove theorems.

And yet, another, could say that mathematicians solve problems.

The first is a defining feature of an even bigger question (what is mathematics?), occurring prior (and hence leading) to axiomatizing.

The second uses axioms to produce facts.

The third is to apply the first two, providing a desired answer to some question.

This post looks at what that question is. What it should be.

Formally, we are asking, what is a good mathematical question?

(We could quip in by saying, is that (and hence this) question a good mathematical question itself?)

Mathematicians spend so much of their time solving questions. Knowledge of what a good question is should help us to solve the question, independent of the question itself.

In fact, this is the point of mathematics; to provide factual statements that are independent of who is providing them.

What is a good mathematical question?

We could say one that is clear and has a purpose i.e. there is no ambiguity to solving $2*2=?$

However, a complicated question can be asked. Life is complicated, often asking questions that seem nonsensical. Therefore, our first question is is not sufficient, certainly it may be a good question, but better can be asked.

A good question can be difficult. Sir Andrew Wiles spent some years proving the Taniyama-Shimura conjecture, which asked several difficult questions.

Some thought it would take a long time to prove as we did not know where to begin to understand the questions let alone to solve it. One man took his time to understand and as a result, we all do.

Sir Andrew Wiles, solving a problem.

Understanding the question is the initial process. You learn the motivation of the question, the required theory. Then, solving the question requires applying theory and mathematical implementation of ideas gained from understanding.

We may conclude with a good question is one that takes some time and careful thinking to solve.

But still, this is not enough. Some complicated questions require fuzzy statements (statements which have not been verified, do not exist and are not true in general and so on). Some complicated questions require numbers which we cannot fathom of.

Some need to be verified by a computer, as we cannot solve that specific part. Some complicated questions themselves are clear yet give way to fuzzy premises in an attempt to solve.

Some complicated questions are just too complicated. You can spend your whole life thinking about them, hopefully solving them. Ideally, a question is a gateway to understanding and gaining knowledge.

Few of us have the time Sir Andrew Wiles or Grigori Perelman did to solve complicated questions.

But these complicated questions are infinitely better than our easy questions. We give an example of easy questions and complicated questions.

Easy questions requires a little bit of time, some specific problem solving skills and a bit of knowledge of undergraduate mathematics.

Complicated questions require a lot of time, new problem solving skills and new mathematical knowledge.

Suppose we want to “solve” the continuum hypothesis. We would need to be comfortable with theories involved in set theory, logic and then learn the method of forcing as outlined by Paul Cohen.

With progress, we would solve several other (easier) questions which would give us more knowledge, independent of solving the original question.

There is a balance, it seems. Is a question asked with some clarity, yet with an interesting element enough?

The Goldbach Conjecture satisfies both of these criterions (clarity; it makes sense, interesting element; knowledge of primes), yet is a surprisingly (actually, unsurprisingly) complicated question, currently unsolved. Although it is of interest, there is no solution, all methods, techniques, ideas and such eventually fail.

Perhaps a good question is one that does not need to be answered. But then, why was it asked? To understand. If we understand, we will (hopefully) eventually solve.

In our modest world, it is fair, then, to say that a good question is one not simple enough to be solved in a couple of minutes; one which requires to carefully think. One requiring some non-trivial theory and to be reasonably solvable.

A question needing different theory to solve is even better.

Say we get asked to find the probability of something event happening, to do so we may need combinatorics to understand our specific event, some theorems regarding the factorial function.

Then we may need linear algebra to solve some linear equations for our distribution, then some knowledge of probability theory, perhaps knowledge of expectation, conditioning probability and such. Through this path, you see why the question was interesting; no clear path existed in solving it.

Eventually you build the intuition to create the clear paths to solve questions.

Such a question is good as you cannot cheat it. You need to think. The solution is not so obvious. A small change to the question can have no answer. But we understand. Then we move to the next question!

Leave a Comment » | Mathematics | Tagged: Continuum Hypothesis, Goldbach Conjecture, Hilbert Problems, mathematics, Probability Theory, Project Euler, Sir Andrew Wiles, Taniyama-Shimura Conjecture | Permalink
Posted by AH

Lecture 1: Sets, Real Numbers, Fields, Ordered Fields

October 30, 2012

The real number system is a set $\{a,b,c, ...\}$ on which the operations of addition and multiplication are defined so that every pair of real numbers (say $a$ and $b$ ) has a unique sum and product, both real numbers, with the following properties:

Given any (and all) $a,b,c \in \mathbb{R}$ , we have

(A) $a+b=b+a$ and $a*b=b*a$ (Commutative laws)

(B) $(a+b)+c=a+(b+c)$ and $(a*b)*c=a*(b*c)$ (Associative laws)

(D) There exist distinct real numbers $0$ and $1$ such that $a+0=a$ and $a*1=a$ , for all $a$ .

(E) For each $a$ there is a real number $-a$ such that $a+(-a)=0$ and if $a \neq 0$ , there is a real number $\frac{1}{a}$ such that $a*\frac{1}{a}=1$

A set on which two operations are defined so as to have properties (A) – (E) is a field.

Motivation: Think about these properties for a second before moving on. (A) establishes what the operations do, (B) then asks what happens when we include more elements, a “double operation” if you like. (C) then mixes the two operands into one. (D) and (E) are closely related. Suppose we want to add an element to our given real number that… gives us our real number? So we are adding nothing, or $0$ . Suppose we have the same idea for our other operand, multiplication, we have the real number $1$ to do this for us.

Then we ask, can we find real numbers that give us what we just discovered, namely $0$ and $1$ from addition and multiplication? For that we have $-a$ and $\frac{1}{a}$ with $a \neq 0$ for the second condition.

Exercise: Can you think of another example of a field?
Exercise: Do the set of integers $\mathbb{Z}$ form a field? Verify through (A) – (E).

The real number system is ordered by the relation $<;$ , which has the following properties:

(F) For each pair of real numbers $a$ and $b$ , exactly one of the following is true: $a <; b$ , $a=b$ , $b<;a$ .

(G) If $a<;b$ and $b<;c$ then $a<;c$ . (Transitive)

(H) If $a<;b$ , then $a+c<;b+c$ for any $c$ and if $b<;c$ then $a*c<;b*c$ .

A field with an order relation satisfying (F) – (H) is an ordered field. The real numbers (and rationals) form an ordered field.

Suppose we get bored of the real numbers and define a new field satisfying everything $\mathbb{R}$ does but is defined as such. For any $a,b \in \mathbb{R}$ we define a new number $z$ as $z = a+b*\imath$ , where $\imath^2 = -1$ . We call this set $\mathbb{C}$ , the set of complex numbers.

Exercise: Verify that $\mathbb{C}$ is a field.
Exercise: Is $\mathbb{C}$ an ordered field? (Use (F) – (H) to prove order or give a counterexample)
(Hard) Exercise: Why have we defined $\imath^2 = -1$ and not $\imath=\sqrt{-1}$ ?

Questions to think about:

Are all fields ordered fields?
Are finite fields ordered?
What do these two above questions (if and when answered) say about ordered fields? (Hint: Infinite number of…)

Leave a Comment » | An Introduction To Real Analysis, Mathematics | Tagged: analysis, mathematics, real, real analysis | Permalink
Posted by AH

Arbias Hashani

Why “Mean” And “Variance” Are So Important

Transformers 3 Puzzle

What is a good mathematical question?

Lecture 1: Sets, Real Numbers, Fields, Ordered Fields

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