gregh1983 and chrisamaphone provided some delicious pasta and cheeses and things over in squirrel hill, and wjl brought Rummikub, which cannot — well, should not — be eaten. Still, it is fun to play. In the end gregh1983 pwned us all after spending the entire game apparently unable to meld his tiles enough.
Like Portal, this flash game has a simple gimmick to make possible something not ordinarily possible, which turns out to be an extremely cute game mechanic. It lacks the nice graphics and writing, of course, but whatever.
Reddit's front page had a link to this image complaining about the liberal application of "and trivially we get" in WP's proof of the central limit theorem by characteristic functions. While I do agree that the article could probably be improved (and I don't think I could personally, else I would) there is a really, really cute proof lurking behind those words that I get the gist of, at least. Here is a non-encyclopedic-style summary:
- What are characteristic functions? They're like a Fourier transform of the probability density function. If the pdf of a random variable X is f, then its char. func. is φX(t) = ∫eitxf(x) dx
- They have the exponentialish property (uncoincidentally reminiscent of how the Fourier transform turns convolution into pointwise multiplication) that if X and Y are independent random variables, then φX+Y(t) = φX(t)φY(t)
- The first few terms of the Taylor series of the characteristic function of a random variable can be worked out just from its moments about zero: consider the defining formula, differentiate n times wrt t, and set t to 0. But for a factor of n! this is the nth derivative, and staring you in the face is a moment integral in∫xnf(x) dx
- In particular, if your mean is zero, and your standard deviation is one, your char. func.'s Taylor series starts off 1 - t2/2 + ...
- If you have n of these dudes all independent, then adding them all up yields a n-way product of their characteristic functions, which are all identical, so it's just an n-th power.
- After some normalization magic, this winds up resembling the formula (1 - t2/n)n
- So, dig deep and remember the limiting formula for the exponential, and realize the char. func. of the result is e-t2. This so happens to be that of the standard normal, QED