## Archive for February, 2008

### Our Topological Brethren

What’s it called when a little donut passes through a bigger donut?

Digestion.

### Genius

Western creationists are freaking Einsteins compared to this blowhard:

### 4 AM Math Post Anyone?

Suppose two random variables $X$ and $Y$ have a discrete joint distribution $p(x,y) = \frac{{C }}{{x!(y - x)!}}$, for some real constant C. Suppose we want to find the moment-generating function for this distribution when $y = 0,1,2,...$ and $x = 0,1,2,...,y$.

By definition, the MGF of this distribution is $M(t_1 ,t_2 ) = E(e^{t_1 x} e^{t_2 y})$, where the function $E$ represents the expected value. Since its a discrete distribution, we write $M$ as:

$C \cdot \sum\limits_{y = 0}^\infty {\sum\limits_{x = 0}^y {\frac{{e^{t_1 x} e^{t_2 y}}}{{x!(y - x)!}}} }$ $= C \cdot \sum\limits_{y = 0}^\infty {\frac{1}{{y!}}\sum\limits_{x = 0}^y {\frac{{y!e^{t_1 x} e^{t_2 y} }}{{x!(y - x)!}}} }$

$= C \cdot \sum\limits_{y = 0}^\infty {\frac{1}{{y!}}\sum\limits_{x = 0}^y {\left( {\begin{array}{*{20}c} y \\ x \\ \end{array} } \right)e^{t_1 x} e^{t_2 y} } }$ … by the definition of the binomial coefficient.

Multiply through by $\frac{{e^{t_2 x} }}{{e^{t_2 x} }}$ and rearrange terms to get:

$C \cdot \sum\limits_{y = 0}^\infty {\frac{1}{{y!}}\sum\limits_{x = 0}^y {\left( {\begin{array}{*{20}c} y \\ x \\ \end{array} } \right)\left( {e^{t_1 + t_2 } } \right)^x \left( {e^{t_2 } } \right)^{y - x}}}$

The inner summation reduces nicely by the binomial formula and we get:

$C \cdot \sum\limits_{y = 0}^\infty {\frac{{\left( {e^{t_1 }e^{t_2 } + e^{t_2}}\right)^y}}{{y!}}}$

By the Taylor series expansion for the exponential function, we can rewrite this as:

$C \cdot \exp (e^{t_1}e^{t_2} + e^{t_2 } )$

Isn’t multiplying by one such an awesome little trick?

### Wounds Are Designed for Bleeding

Mike Dunford over at The Questionable Authority writes about the rejection of the Discovery Princetitute blog from ResearchBlogging.org. He made one excellent point (among many) that deserves to be fondled lovingly. In a totally platonic way, of course. Unless you’re parents aren’t around. Then it’s totally cool. Uh.. anyway.

He writes:

The scientific community’s distaste for the Discovery Institute isn’t caused by intolerance for dissenting views. It’s the effect of the years that the Discovery Institute has spent publicly attacking science and scientists.

Well, this just sums up the problem with the ID movement, and on multiple fronts. This misconception, this idea that everything has an agenda, a clear, directed creative purpose behind it is just factually wrong. It’s obviously wrong. It’s wrong when it comes to evolution. It’s wrong when it comes to criticisms of ID. It’s wrong when you’re talking almost everything. The weather has no agenda behind it (although I suppose most creationists would believe it does). An ant dying accidentally under my boot has no agenda behind it. Follicle bacteria imbibing your dermal sebum has no agenda behind it. “Agenda” simply isn’t a prominent feature of our universe, numerically-speaking. Even if you believe in a prime mover, that’s only ONE process out of the infinitely many processes which has an agenda. I mean, come on.. our beloved, oh-so-perfect species can barely keep its head straight on most occasions. It’s only through GRADUAL, PROGRESSIVE processes that society is enhanced, not perfunctory moral crusades.

Just put the science down for a minute. You’re out of your league. Take a look at your assumptions first. If you want to be a scientist, that’s what you have to do. Once you’ve taken a nice long pilgrimage, oh prodigal son, you can come back and apologize for your wrongdoings. And then you can do all of the cool stuff that scientists do. Peer pressure.. PEER PRESSUREEEE.. DON’T YOU WANT TO BE COOL LIKE US?

😛

### The Copyright Kerfluffle @ ResearchBlogging

The blogodome has been all in a kerfluffle over the latest Perfunctory Institute episode involving Casey Luskin and his crow-eating behavior concerning copyright infringement. Specifically, Luskie apparently took an image off of ResearchBlogging (or his “friend gave it to him”) without permission and he used it to “web log” about some essay in some biology journal (DI people don’t blog, they “web log”–it’s like blogging, only in the style of ten years ago. Y’know, without all the nifty commenting and RSS feeding and whatnot.. all the products of blogging modernity and so forth. Although they do have trackbacks–which speaks to the media whority of the DI. Digressions!).

### A Special Case

My statistics professor told a joke today that, while relatively funny, is also a bit interesting too. It goes like this:

A math professor is proving some general theorem, and ends up with a result involving $x$‘s and $y$‘s. One student raises his hand and asks, “Can you show us a special case of that theorem?”. In other words, the student wanted a specific application of the theorem. The professor proceeds to erase the $x$‘s and $y$‘s and replace them with $x_0$‘s and $y_0$‘s. The student replies, “Ah! That makes more sense.”

### Fracking Fractals

I found this amusing description while scouring my feedreader today:

# fractal wrongness

The state of being wrong at every conceivable scale of resolution. That is, from a distance, a fractally wrong person’s worldview is incorrect; and furthermore, if you zoom in on any small part of that person’s worldview, that part is just as wrong as the whole worldview.

Debating with a person who is fractally wrong leads to infinite regress, as every refutation you make of that person’s opinions will lead to a rejoinder, full of half-truths, leaps of logic, and outright lies, that requires just as much refutation to debunk as the first one. It is as impossible to convince a fractally wrong person of anything as it is to walk around the edge of the Mandelbrot set in finite time.

If you ever get embroiled in a discussion with a fractally wrong person on the Internet–in mailing lists, newsgroups, or website forums–your best bet is to say your piece once and ignore any replies, thus saving yourself time.