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From the article: "Both distributions below have standard deviations of 1" I thought fat tailed distributions don't have a variance. But apparently I'm using the term in a stricter sense than other people. See http://en.wikipedia.org/wiki/Fat-tailed_distribution#Definit... for details if you were wondering, too.
Earlier quoted context omitted.
yeah i gave myself some grief about that aspect but ultimately decided different scales was more misleading. could go either way. also, the kurtosis of a normal distribution is 3, not 0.
Many statistical packages (SAS IIRC) subtract 3 from the kurtosis to set a normal distribution at 0. This is an infuriating tendency.
the [edit: outer envelope of the ] walk should slowly deviate from zero, shouldn't it (the variance of the sum increases with time / a random walk is a walk )? what i am seeing is returning to zero much more strongly than i would have expected. what is the prng that you are using? i suspect it's not that great.
Earlier quoted context omitted.
yeah i gave myself some grief about that aspect but ultimately decided different scales was more misleading. could go either way. also, the kurtosis of a normal distribution is 3, not 0.
Yes, on your scale it appears that is the case. Most definitions I've seen have a -3 constant to zero out the normal, but if you say that's weird I won't argue. Incidentally, why didn't you extend the kurtosis scale to allow for a platykurtic distribution? EDIT: to the original question, I'm not suggesting the distributions should have different scales, but rather that the scale of both should be static.
This is great, but I have a slight complaint. Sliding the kurtosis indicator changes the left distribution, which makes sense. However, it also changes the appearance of the right, normal distribution, which is misleading. Normal distributions have [EDIT: constant] kurtosis. I realize that the appearance is changing because the scale is changing so that the max is always pegged. However, it might be less confusing if…
the [edit: outer envelope of the ] walk should slowly deviate from zero, shouldn't it (the variance of the sum increases with time / a random walk is a walk )? what i am seeing is returning to zero much more strongly than i would have expected. what is the prng that you are using? i suspect it's not that great.
Symmetric random walks, including those with step sizes drawn from a Gaussian process with mean zero, have expectation 0 at any time t. Since there's no drift term, either walk won't be expected to slowly deviate from zero.
the [edit: outer envelope of the ] walk should slowly deviate from zero, shouldn't it (the variance of the sum increases with time / a random walk is a walk )? what i am seeing is returning to zero much more strongly than i would have expected. what is the prng that you are using? i suspect it's not that great.
Symmetric random walks, including those with step sizes drawn from a Gaussian process with mean zero, have expectation 0 at any time t. Since there's no drift term, either walk won't be expected to slowly deviate from zero.
But the variance of your probability distribution of where you'll be at time t is linear in t. So say that your variance is v(t)=t. Then at t=1, there is a 32% chance that you'll be outside of the range (-1,1). As you can see, as t increases, you expected to drift further than further.
So while the expectation of x(t) may be 0 for all time, the expectation of |x(t)| scales like sqrt(t) (the standard deviation of the distribution).
The thing that can't be easily seen in pictures is that exponential distributions move differently than gaussian distributions. When the variance increases for a gaussian, it flattens out. When the variance increases for an exponential, the whole thing spikes out to the right, and the area under the tail actually increases. It really screwed things up until I started treating it for what it really was.
I think the biggest fat tail distribution of all is the things that developers are asked to do on a day to day basis. The fat tail is where frameworks break down, where development processes and procedures break down. Most of the stuff I do is stuff that I have never done before and will never do again.