Central Limit Theorem

The Central Limit Theorem says that the sum of enough identically distributed independent random variables looks like a normal distribution. Testing that via simulation. Particles follow a (simulated) brownian motion from the top center of the screen and have their point of impact recorded at the bottom. Each particle's path consists of 18000 events, whereby, at each, it moves vertically "-dy" and horizontally either "dx" or "-dx", with equal probability. As the particle reaches the bottom of the screen, a special impact event is recorded. The distribution of these impact events is, by construction, the distribution of the sum of the random variables giving the horizontal position changes at each time step. Note how the impact distribution is similar to the predicted gaussian.

http://computability.blogspot.com

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• random
• walk
• finance
• science
• math
• mathematics
• education
• brownian
• motion

[benschaeffer]

In the video, the balls are scattering because the simulation makes them :-) Basically at every downward step, each ball makes a random decision about whether to go right or left. Over time, the accumulated differences in random decisions result in very different positions in space! This same effect occurs in real life in various physical devices people have created to demonstrate this mathematical truth. In those devices, the falling balls tend to impact small obstacles like pins.

[Booredyeah]

How is this meaningful in anyway whatsoever, all the balls are being droped in from the center, it seems logical that there is a higher probability of the balls landing in the center, no?

[benschaeffer]

Thanks for the comment! I agree it is obvious that the balls will tend to land in the center. The mathematics predicts more, however... namely that the resulting distribution will have a particular shape (namely that of a gaussian).

[bettiethebadger]

if you sample until infinity (limit n tends to inf) why doesn't the distribution degenerate to a point mass since we have sampled the entire population so the sample mean should equal the population mean parameter with no variance?

[benschaeffer]

Yes, there is convergence as n goes to infinity. But instead of the distribution degenerating to a point, it converges to a normal distribution as n increases (which is the content of the Central Limit Theorem). For instance, the proportion of paths falling into one of the buckets gets closer and closer (in the sense of probability, i.e. that large deviations from the expected value are unlikely) to that predicted by the normal distribution (shown in the video by the blue line).

[VideoJunkei]

The Seattle Science Center had a display of this for YEARS 70's to 90's....it used real balls, and was 15-20 feet high. The balls were taken via toothed scoops on chain drive on the left side to the top, and dumped in the center funnel, then went through a field of pegs, then the tall catch boxes, as above.

[benschaeffer]

Yes, I got the idea from one of my favorite exhibits at the Chicago Museum of Science and Industry, which was exactly as you describe! Guess that many science museums had similar displays. It's amazing that such a deep theorem of abstract mathematics has a physical demonstration.

[onionpizza]

I counted 2,935,126,274,643,951 balls in this experiment. Anyone got a different count?

[Amishman35]

REMEMBER: You're unique, just like everyone else.

[lovelplants]

it's basically  distribution.. it'll apply to statistics

[machincuisi]

Y ESAS MAMADAS QUE PUTO GRINGOO!!!!

[boxedbocchan]

Let's see if I can get this comment seen in my statistics class.

[Suckittrebek5252]

The only reason I came here is because Roland Good III told me too!

[RayWillJohnson]

Nothing random about the results

[bosshoss502]

Thought I saw purple Jaws with blue teeth... never-mind... carry on.

[slonopochron3000]

wait, but is this really a demonstration of the central limit theorem? isn't it just a demonstration of the fact that the final location of a random walk is normally distributed? i mean you're basically just applying the law of large numbers to a random walk, not SUMMING the random walks. knowwhatimean?

[slonopochron3000]

this is amazing.

[2002budokan]

It is a wonderfull mnemonic to remember the central limit theorem, I'll never forget it after watching this. But this should be accellarated. at least 5 times. One minute video would be optimal.

[snuchia]

@Booredyeah Yes, that's why the center-of-mass of the distribution is directly under the point where the particles enter the frame. But the details of the shape of the curve require some deeper reasoning.