Why is the normal distribution not normal? - page 33
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Therefore, we have an observable hybrid.
The hybrid is quite harmonious - if we ignore non-stationarity of the process generating this distribution. The most important thing is that it is stable (its integral is very similar to the fractal Brownian motion that Peters writes about in his "Fractal Analysis of Financial Markets"). What is the stability of the distribution, I hope you remember?
The hybrid is quite harmonious - if we ignore non-stationarity of the process generating this distribution. The most important thing is that it is stable (its integral is very similar to the fractal Brownian motion that Peters writes about in his "Fractal Analysis of Financial Markets"). What is distributional stability, I hope you remember?
I have no idea about the formal definition of sustainability, so spit it out! ;)
About intuitive - the harmonicity and stability of this fractal I warmly approve and hopefully understand quite well.
Roughly speaking, robustness is when the distribution of the sum of two independent equally distributed variables (possibly with different parameters) has a distribution F as well. Stable is normal (expectation and variance are summed), Cauchy, uniform and a bunch of others.
What is the sum implied here? Algebraic? That is, we have two generators, working on the same distribution (possibly with different parameters). At each step each generates one value: x and y. Then the sum is a random variable z=x+y. So ?
Right, we're not talking about processes, we're talking about distributions.
Roughly speaking, robustness is when the distribution of the sum of two independent equally distributed quantities (possibly with different parameters) has the same distribution as F. Stable is normal (expectation and variance are summed), Cauchy, uniform and a bunch of others.
I am not out of the blue surprised. Always thought that only normal can have this property, and that this is its essence. And all others (except for uniform at infinity) tend to normal when summed up. Is there no error? Aren't you being too harsh?
I don't think it's too much.
If Z = X + Y, then pdf Z is the convolution of pdf X and pdf Y. You want to practice with Cauchy, remember your youth.
Here's another look at the Other properties. It explicitly says that it's stable. But the definition of stability in the link is very different, contrived... But even there we can clearly see that there are many different stable distributions anyway.
Here's another look at the Other properties. It explicitly says that it is stable. However, the definition of stability in the link is very different, contrived... But even there you can clearly see that there are many different stable distributions anyway.
Stable distributions are not many, there is one. The normal, Cauchy and Levy distributions are the three famous special cases of the stable distribution, there are no others - https://en.wikipedia.org/wiki/Stable_distribution
In English, they are called stable distributions. Google brings up a lot of links. The most interesting is this one http://fs2.american.edu/jpnolan/www/stable/stable.html
I'm shocked. According to this logic, the first differences from a Cauchy distribution also generate a Cauchy distribution. The second (differences from the first differences) are also coshy. The third ones are also coshy. And so on.
It doesn't make sense to me. I always thought that any input distribution with such consecutive taking of "prizes" will inevitably quickly reduce to normal. Should I go get drunk...? :) Nah. I'd better check it tomorrow. I'll write a script and check it.
I'm shocked. According to this logic, the first differences from a Cauchy distribution also generate a Cauchy distribution. The second ones (the differences from the first differences) are also Coshi. The third ones are also coshy. And so on.
It doesn't make sense to me. I've always thought that any input distribution would inevitably quickly reduce to a normal distribution by taking the "prizes" so consistently.
Yep, there you have it, the pleasant surprise of fat-tailed distributions.
And, best of all, even the sample average from Cauchy is distributed according to exactly the same Cauchy.
By the way, the standard normal is not so nasty at all, but white and fluffy: the s.c.a. of the sample mean decreases as the sample size increases.