# How may I generate a distribution which follows power law?

I would like to generate a distribution which follows a power law in a rather peculiar way. I have a lot of marbles, I take one of them at a time and put in a set with a certain distribution of probability: let's say that when I take marble N there is a probability $q(N)$ that it will be put in a new set, and a probability $q(S_k)$ to end up in set $k$, depending on the number of marbles already present in the set. Naïvely, I would say that the probability is linear with respect to the number of marbles in the sets, at least in the simple case where $\alpha$ is 2 in the overall distribution $p(x)=(\alpha-1)x^{-\alpha}$, but I am not sure about this; neither I am sure about which function is $q(N)$ (maybe $1/N$?).

Is there an easy way to come up with such a distribution?

• I think your title is a bit misleading; I was expecting this to be a question about generating random variables according to a power law distribution on a computer (where we are free to come up with how to do it). At any rate your problem seems to be underspecified: what distribution do you want to get at the end of this procedure?
– Ian
Mar 6, 2017 at 12:51
• @Ian: feel free to modify the title to something more clear! What I would like to end with is a (discrete) distribution where if I rank the sets of marbles in decreasing size I roughy have that set #$n$ has $1/n$ of marbles of set #$1$. Something like Galton machine, but with a distribution $1/x$ instead of binomial.
– mau
Mar 6, 2017 at 22:15