Category Archive

You are currently browsing the category archive for the ‘General’ category.

ST 1021: Question about Expectation

November 19, 2011 in General, Grinds | Leave a comment

Consider the following question. X is supposed to represent the sale price of a hotel room, while Y represents the cost price. Therefore the profit is given by X-Y. I am going to use the term expected average as opposed to the more standard expected value or expectation. 

There is a problem with the interpretation and I wouldn’t treat this particular exercise with much importance.

Suppose that X and Y are independent random variables with distributions

 

Find the expected average of the profit on a single room. Find the expected average of the profit on 1,000 rooms. Find the probability that the profit on 1,000 rooms is less than 20,000.

Solution : The expected average of a variable is given by:

\mathbb{E}[X]=\sum_ix_i\mathbb{P}[X=x_i]=\sum_ix_ip_i.

Now expected average is linear:

\mathbb{E}[X+\lambda Y]=\sum_i(x_i+\lambda y_i)p_i=\sum_{i}x_ip_i+\lambda\sum_i y_i p_i

=\mathbb{E}[X]+\lambda\mathbb{E}[Y].

Read the rest of this entry »

Project Maths: A More Geometric Approach to De Moivre’s Theorem and Complex Roots

November 19, 2011 in General, Grinds, Leaving Cert, MA 1008, MS 3011 | 3 comments

Introduction

This is just a short note to provide an alternative way of proving and using De Moivre’s Theorem. It is inspired by the fact that the geometric multiplication of complex numbers appeared on the Leaving Cert Project Maths paper (even though it isn’t on the syllabus — lol). It assumes familiarity with the basic properties of the complex numbers.

Complex Numbers

Arguably, the complex numbers arose as a way to find the roots of all polynomial functions. A polynomial function is a function that is a sum of powers of x. For example, q(x)=x^2-x-6 is a polynomial. The highest non-zero power of a polynomial is called it’s degree. Ordinarily at LC level we consider polynomials where the multiples of x — the coefficients — are real numbers, but a lot of the theory holds when the coefficients are complex numbers (note that the Conjugate Root Theorem only holds when the coefficients are real). Here we won’t say anything about the coefficients and just call them numbers.

Definition

Let a_n,\,a_{n-1},\,\dots,\,a_1,\,a_0 be numbers such that a_n\neq 0. Then

p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,

is a polynomial of degree n.

In many instances, the first thing we want to know about a polynomial is what are its roots. The roots of a polynomial are the inputs x such that the output p(x)=0.

Read the rest of this entry »

Proof of the Cauchy-Schwarz Inequality for Euclidean Spaces

November 2, 2011 in General, Grinds | Leave a comment

Theorem: Cauchy-Schwarz Inequality

Let a_1,a_2,\dots,a_n and b_1,b_2,\dots,b_n be sequences of real numbers. Then we have

\left|\sum_{i=1}^na_ib_i\right|\leq\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum_{i=1}^nb_i^2}.

Proof : Consider the following quadratic function f:\mathbb{R}\rightarrow\mathbb{R}:

f(x)=\sum_{i=1}^n(a_ix+b_i)^2.

Note at this point that f(x)\geq0 for all x\in\mathbb{R}.

f(x)=\sum_{i=1}^n(a_i^2x^2+2a_ib_ix+b_i^2)

=\left(\sum_{i=1}^na_i^2\right)x^2+\left(2\sum_{i=1}^na_ib_i\right)x+\sum_{i=1}^nb_i^2.

That is f is a \bigcup or `+x^2‘ positive quadratic so has one or no roots. That means the roots are real and repeated or complex so that we have b^2-4ac\leq 0 where f(x)=ax^2+bx+c:

\left(2\sum_{i=1}^na_ib_i\right)^2-4\left(\sum_{i=1}^na_i^2\right)\left(\sum_{i=1}^nb_i^2\right)\leq0

\Rightarrow \left(\sum_{i=1}^na_ib_i\right)^2\leq \left(\sum_{i=1}^na_i^2\right)\left(\sum_{i=1}^nb_i^2\right)

Now take square roots (remembering \sqrt{x^2}=|x|.) \bullet

Leaving Cert Project Maths: Proofs for 2012

October 25, 2011 in General, Grinds, Leaving Cert | 19 comments

With the new Project Maths programme being developed as we speak, diligent students might like to know which proofs are examinable under the new syllabus so they know which to look at.

It can be difficult to sift through the syllabi at projectmaths.ie but I have gone through them and here are the proofs required.

The Mathematics of Card Shuffling

September 19, 2011 in General | Leave a comment

I’m giving a talk in Blackrock Castle Observatory on October 7. See the link for more details.

http://www.bco.ie/2011/09/first-fridays-at-the-castle-celebrating-50-years-of-human-spaceflight/

Leaving Cert Project Maths: The Average Vs The Expected Value

March 29, 2011 in General, Grinds, Leaving Cert | Leave a comment

The Average

The average or the mean of a finite set of numbers is, well, the average. For example, the average of the (multiset of) numbers \{2,3,4,4,5,7,11,12\} is given by:

\text{average}=\frac{2+3+4+4+5+7+11+12}{8}=\frac{48}{8}=6.

When we have some real-valued variable (a variable with real number values), for example the heights of the students in a class, that we know all about — i.e. we have the data or statistics of the variable — we can define it’s average or mean.

Definition

Let x be a real-valued variable with data \{x_1,x_2,\dots,x_n\}. The average or mean of x, denoted by \bar{x} is defined by:

\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}=\frac{\sum_{i=1}^nx_i}{n}.

Read the rest of this entry »

Leaving Cert Maths: Differentiation from First Principles

March 29, 2011 in General, Grinds, Leaving Cert | 1 comment

In Leaving Cert Maths we are often asked to differentiate from first principles. This means that we must use the definition of the derivative — which was defined by Newton/ Leibniz — the principles underpinning this definition are these first principles. You can follow the argument at the start of Chapter 8 of these notes:

https://jpmccarthymaths.com/wp-content/uploads/2010/07/lecture-notes.pdf,

to see where this definition comes from, namely:

f'(x)\equiv \frac{dy}{dx}=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}. (*)

Read the rest of this entry »

My Understanding of Non-Commutative Geometry

March 14, 2011 in General | 1 comment

This is intended to be the subject of a short postgraduate talk in UCC. At times there will be little attempt at rigour — mostly I am just concerned with ideas, motivation and giving a flavour of the philosophy. Also it is fully possible that I have got it completely wrong in my interpretation!

Introduction

It is a theme in mathematics that geometry and algebra are dual:

\text{Geometry }\leftrightarrow \text{ Algebra}

Arguably this theme began when Descartes began to answer questions about synthetic geometry using the (largely) algebraic methods of coordinate geometry. Since then this duality has been extended and refined to consider:

\text{Spaces }\leftrightarrow \text{ Algebra of Functions on the Space}

Here a space is a set of points with some additional structure, and the idea is that for a given space, there will be a canonical algebra of functions on the space. For example, given a compact, Hausdorff topological space X, the canonical algebra of functions is C(X) — the continuous functions on X.

Read the rest of this entry »

The Fundamental Theorem of Linear Programming

February 5, 2011 in General | Leave a comment

Suppose we want to maximise (or minimise) the linear function C(x,y)=\lambda x+\mu y on a set S. Suppose S is defined as the solution of  the system of three linear inequalities:

a_1x+b_1y\leq c_1

a_2x+b_2y\leq c_2

a_3x+b_3y\leq c_3

In general, the solution set of these inequalities will be a triangular sea of x and y:

The points \mathbf{a}, \mathbf{b}, \mathbf{c} are the extreme points on S. We want to prove that C(x,y) is maximised at \mathbf{a}, \mathbf{b} \text{ or } \mathbf{c}.

Read the rest of this entry »

Leaving Cert: Integration

November 22, 2010 in General, Grinds, Leaving Cert, MA 1008 | Leave a comment

A short note covering integration for Leaving Cert maths.

 

(Please note that the proof of the Fundamental Theorem of Calculus inside isn’t quite correct. We need the Mean Value Theorem to prove it but the one in here is just for illustrative purposes.)