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The math that explains why bell curves are everywhere

quantamagazine.org

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Re: The math that explains why bell curves are everywhere

#61

Post by abetusk »

tsunamifury wrote: post #25

Bell curves are everywhere because all distributions of any properties clump in some way at some level. The basics of any probability shows this. The result is you “seeing” bell curves everywhere. Aka clumps. This is a tautology to the extreme.

No, that's not true.

If sums of independent identically distributed random variables converge to a distribution, they converge to a Levy stable distribution [0]. Tails of the Levy stable distribution are power law, which makes them not Gaussian.

[0] https://en.wikipedia.org/wiki/Stable_distribution

Re: The math that explains why bell curves are everywhere

#62

Post by smcin »

hilliardfarmer wrote: post #42
bicepjai wrote: post #40

This is one of my favorite philosophical questions to ponder. I always ask it in interviews as a warmup to get their thoughts. I’ve noticed that interviewees often curl up, thinking it’s a technical question, so I’ve been modifying the question one after the other to make it less scary. The interviews are for data scientist roles.

A lot of times I can't tell if I'm the idiot or if everyone else is. Says that this isn't an interesting question at all and the article was horrible. I studied data science for a few years but I'm no expert, but it seems pretty obvious to me that if you make a series of 50/50 choices randomly, that's the shape you end up with and there's really nothing more interesting about it than that.

Sampling 50/50 choices would be a binary distribution that (very crudely) approximates a normal distribution.

But the counterintuitive thing about the CLT is that it applies to distributions that are not normal.

Re: The math that explains why bell curves are everywhere

#64

Post by woopsn »

abetusk wrote: post #32

Sorry, does the article actually give reasons why the bell curve is "everywhere"? For simplicity, take N identically distributed random variables that are uniform on the interval from [-1/2,1/2], so the probability distribution function, f(x), on the interval from [-1/2,1/2] is 1. The Fourier transform of f(x), F(w), is essentially sin(w)/w. Taking only the first few terms of the Taylor expansion, ignoring constants,…

There's a paragraph on discovery that multinomial distributions are normal in the limit. The turn from there to CLT is not great, but that's a standard way to introduce normal distributions and explains a myriad of statistics.

Re: The math that explains why bell curves are everywhere

#65

Post by causalityltd »

Causes mostly add up: molecular kinetic energies aggregate to temperature, collisions to pressure, imperfections to measurement errors, etc. So, normal or CLT is the attractor state for the unexceptional world.

BUT for the exceptional world, causes multiply or cascade: earthquake magnitudes, network connectivity, etc. So, you get log-normal or fat-tailed.

Re: The math that explains why bell curves are everywhere

#66

Post by throwaway81523 »

DroneBetter wrote: post #2

I hate Quanta a lot a vast amount of fluff for less than a college statistics professor would (hopefully) be able to impart with a chalkboard in 10 minutes, when Quanta has the ability to prepare animated diagrams like 3Blue1Brown but chooses not to use it they could go down myriad paths, like how it provides that random walks on square lattices are asymptotically isotropic, or give any other simple easy-to-understan…

Quanta used to have tons of good stuff and not much crap. Now there's enough crap that if there's still good stuff, it gets lost in the noise.

Re: The math that explains why bell curves are everywhere

#68

Post by rtgfhyuj »

fritzo wrote: post #7

Hot take: bell curves are everywhere exactly because the math is simple. The causal chain is: the math is simple -> teachers teach simple things -> students learn what they're taught -> we see the world in terms of concepts we've learned. The central limit theorem generalizes beyond simple math to hard math: Levy alpha stable distributions when variance is not finite, the Fisher-Tippett-Gnedenko theorem and Gumbel/Fr…

any good resources to understand more about them?

Re: The math that explains why bell curves are everywhere

#69

Post by pfortuny »

gowld wrote: post #23
sobellian wrote: post #9

Earlier quoted context omitted.

IIRC there's a video by 3b1b that talks about that, and it is important that gaussians are closed under convolution.

That makes it an equilibrium point in function space, but the other half is why it's an a global attractor.

There must be a contractive nature in "passing to the limit". And then Brower's fixed point theorem.

(I know it is very easy to do "maths" this way).

Re: The math that explains why bell curves are everywhere

#70

Post by btilly »

mikrl wrote: post #3

Great article. Personally I have been learning more about the mathematics of beyond-CLT scenarios (fat tails, infinite variance etc) The great philosophical question is why CLT applies so universally. The article explains it well as a consequence of the averaging process. Alternatively, I’ve read that natural processes tend to exhibit Gaussian behaviour because there is a tendency towards equilibrium: forces, homeost…

The key principle is that you get CLT when a bunch of random factors add. Which happens in lots of places.

In finance, the effects of random factors tend to multiply. So you get a log-normal curve.

As Taleb points out, though, the underlying assumptions behind log-normal break in large market movements. Because in large movements, things that were uncorrelated, become correlated. Resulting in fat tails, where extreme combinations of events (aka "black swans") become far more likely than naively expected.

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