In which I learn the point of measure theory

January 26, 2009 at 5:52 pm | In Mathematics | 5 Comments

In an earlier post I showed the engineer’s way of doing a calculation. This is the mathematician’s way (I think, although with notation halfway between probability theory and measure theory):

Define \Omega = \{\pm1\} \times \mathbb{R}. Then define a measure on this space by d\mu = (dp_{-1}+dp_{1})/2 \times \frac{1}{\sigma\sqrt{2\pi}}\exp(-z^2/2\sigma^2) dz where dp_a is the point measure at a. This is a probability space because \mu(\Omega)=1.

Our random variable is now Y \colon \Omega \to \mathbb{R} given by Y(x,z) = x + z. Then F(t) = P(Y \leq t) = \mu\{x+z \leq t\} = \mu\{x=-1,z \leq t+1\} + \mu\{x=1, z \leq t-1\}. So F(t) = \frac{1}{2\sigma\sqrt{2\pi}}(\int_{-\infty}^{t+1}\exp(-z^2/2\sigma^2) dz+\int_{-\infty}^{t-1}\exp(-z^2/2\sigma^2)dz). Then f(t)=F'(t) = \frac{1}{2\sigma\sqrt{2\pi}}(\exp(-(t+1)^2/2\sigma)+\exp(-(t-1)^2/2\sigma) just the same as in the earlier post.

And there weren’t even any distributions! The moral of the story is that there’s no difference between continuous and discrete – they’re both just measures.

A shorter cricket puzzle

January 24, 2009 at 4:24 pm | In Cricket | 5 Comments

Two batsman are on 94 with 7 runs required to win and 2 balls remaining. How can they both score their centuries?

A cricket puzzle

January 24, 2009 at 4:09 pm | In Cricket | 3 Comments

Have a go at this puzzle from MIT’s puzzle competition.

I’m close to a solution but I just can’t get Mike Hussey facing at the start of Patel’s overs like I need to. Frustrating…

A calculation I don’t really understand

January 20, 2009 at 8:17 pm | In Mathematics, Uni | 5 Comments

We’re trying to model interference on a communication channel. The line noise is normal with mean 0 and variance \sigma so has probability density function (pdf) f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-x^2/2\sigma^2}.

We have a random variable X that spits outs -1 or 1 with equal probability (this is a model for the coded data that we want to transmit). This has pdf g(x) = (\delta(x+1)+\delta(x-1))/2.

Then the random variable that we get at the other end Y = N+X has p.d.f. given by the convolution h(x) = \int f(y)g(x-y) dy. By doing this integral using properties of the delta function/distribution/measure we get h(x) = \frac{1}{2\sigma\sqrt{2\pi}}(\exp(-(x+1)^2/2\sigma^2)+\exp(-(x-1)^2/2\sigma^2)). This gives quite nice pictures

small

big

which I guess show something important engineeringly: If you can get your line noise down to where the bumps don’t overlap then it’ll be pretty easy to figure out whether a bit was meant to be 1 or -1.

But I don’t really understand the maths. I guess I can accept the facts about pdfs but I’m kind of confused about continuous v discrete random variables. Should I always think of discrete random variables as being delta functiony things when I want to work with them?

World Baseball Classic

January 19, 2009 at 6:23 pm | In Baseball | 3 Comments

The preliminary Australian team for the world baseball classic is here.

There’s a good review of the Australian team here. It looks like we’ve got AA/AAA level players plus major league first base Huber and pitchers Rowland-Smith & Balfour.

Rowland Smith will hopefully start one game. See article here

Balfour wants to be the closer apparently but I’d rather him be used as a relief ace – bring him in in relief at any time the starter’s out and we’re tied, 1 run behind or 1 run ahead.

You can see the schedule here. It’s double elimination which basically means that as long as we beat South Africa (should be easy enough) we go into a one game playoff with either Cuba or Mexico to reach the second round. That’s a tough game but you never know.

Cynical reviews of my courses 3 days into the semester

January 19, 2009 at 12:57 pm | In Mathematics, Uni | 3 Comments

MATH518 – Integral Lattices. Superfast lectures about things I’ve never heard of.

MATH536B – Algebraic Geometry. Superfast lectures that started by saying “Everything we said about varieties last semester is true about schemes, except when it’s not. Let’s keep going”.

ECE637 – Channel Coding. This is an engineering course. We’ve already used the fact that 10^3=2^10 as well as used the word ‘finite’ to mean non-zero.

MATH599 – Independent study. Continuing research from last semester. We’re stuck on a lemma from the paper whose entire proof consists of the word `straightforward’.

Teaching – Calc I. This is one hour a day, 5 days a week. At least most of the homework is online (webassign) so it’s less work for me.

Talk topics

January 19, 2009 at 12:37 pm | In Mathematics, Uni | 2 Comments

I should give a talk on something at the grad colloquium this semester. I don’t really think the stuff I’m researching/learning is very interesting so I’d like to talk about something easier that I can really explain in 50 minutes. But I don’t know what to talk about. Any suggestions? These are some of my ideas:

  • Why you can’t integrate e^{x^2} in terms of elementary functions (differential Galois theory?)
  • mathematics of sport (mainly this would be cricket, which is a bit of a problem)
  • groebner bases (+ geometrical theorem proving?)
  • combinatorial game theory
  • something in logic/set theory

The problem with being on the internet is that people think you know things

January 19, 2009 at 12:30 pm | In Blogging about my blog, Cricket, Mathematics | 6 Comments

An email from some dude who read a project I did (you can see it on my website):

Dear Martin Leslie,
My name is (removed), and I’m a student at Universidad de Buenos Aires, working
on Mental Poker protocols for my thesis.

I’ve read your project on SRA Mental Poker Protocol.

In some place, you state that
“However there do exist more general attacks on SRA and
the fact that it leaks some information suggests that it probably leaks other
information, perhaps not as well known as quadratic residues.”

Could you give me references to the “more general attacks on SRA” ?

Thank you very much,

I have no idea what I meant and replied as such.

Even more bizarre is this comment on my about page.

Hello,
I am contacting you from Sky News. We would like to do an interview with you today regarding Kevin Pietersen resigning as England Captain.

Please can you call me urgently on 0207 585 4570.

Best Jean

what

Blog at WordPress.com. | Theme: Pool by Borja Fernandez.
Entries and comments feeds.