In the example under log-normal, you talk about stock prices. Stock prices are essentially approximately arbitrary, since they depend on the number of shares issued — you can have a stock split, were each existing stock gets replaced with 2 (or more) new ones, making no real difference to the ownership of the company.
Could you illustrate instead with market capitalisation of the companies?
Also, in your discussion of scale invariance, you talk about the size of stars and say “meters are an arbitrary unit”. But that is equally true of metres used to measure people’s heights, which is the example you use for normal distributions. I think that scale invariance means something subtly different from saying that a unit of measure is arbitrary. I think (though am not sure) that it’s more like saying that going from 10 to 20 is just as likely as going from 1 to 2, and the same is true of going from 1000 to 2000, or from 5 million to 10 million. I.e., doublings (or whatever scaling) are equally likely whatever scale you are currently at.
Thanks for posting this.
A couple of nit-picks:
In the example under log-normal, you talk about stock prices. Stock prices are essentially approximately arbitrary, since they depend on the number of shares issued — you can have a stock split, were each existing stock gets replaced with 2 (or more) new ones, making no real difference to the ownership of the company.
Could you illustrate instead with market capitalisation of the companies?
Also, in your discussion of scale invariance, you talk about the size of stars and say “meters are an arbitrary unit”. But that is equally true of metres used to measure people’s heights, which is the example you use for normal distributions. I think that scale invariance means something subtly different from saying that a unit of measure is arbitrary. I think (though am not sure) that it’s more like saying that going from 10 to 20 is just as likely as going from 1 to 2, and the same is true of going from 1000 to 2000, or from 5 million to 10 million. I.e., doublings (or whatever scaling) are equally likely whatever scale you are currently at.