uj_ wrote:
phineasgage wrote:

The continuous distribution from uj_ works great. Although it isn't perfect in the discrete case, it can still be used if the size of N is large and the application doesn't demand perfect accuracy in the lower range.

I haven't been doing calculus for a while but I suddently realized that this,

X = e^[ln(N)*U]

can be further reduced to,

X = [e^[ln(N)]]^U = N^U

This means that the pow method can be used for the simulation code, like

```
Random rand = new Random();
int n = 100;
double dU = rand.nextDouble();
double dX = Math.pow(n, dU); // dX is the random number on the distribution
```

Considering that this distribution is so easy to simulate (in the continous case) it's quite astonishing that there are so few references to it. I haven't found a single one.

And the distribution is interesting. Every number in the range 1 to N comes up with a probability that's proportional to the inverse of the number. Informally this corresponds to the notion that "the shorter/smaller/sooner the more probable, the longer/larger/later the less probable". This should be a common situation in "nature". Doesn't stuff tend to split up in many small and few large? Doesn't people prefer sooner rather than later?

Further the distribution has a "fat tail". This means that numbers far away from the average still has a substantial probability of occuring. This is not the case in the Gaussian/Normal distribution for example where probabilities quickly get extremely small (like not even once in the lifetime of the universe).

The distribution also is mathematically truncated. It's limited to a specific range. Again, this is not the case with the Gaussian/Normal distribution. It must be artificially truncated in principle skewing it.

Now what should this good cigarr be called? Well I suggest the Truncated Harmonic Distribution until someone manages to Google up its given name -:)

I was surprised as well not to find a discrete version of this distribution, at least somewhere. Kleinberg's results are interesting, that in small world networks the optimal random link distribution is 1/d^r, where d is the distance between the nodes, and r is effectively the number of dimensions in the network. The r=1 case is interesting in how it breaks calculations for existing power law distributions, probably because integrating 1/x is a special case from the general 1/x^r(?) The Harmonic Series is also interesting in that if you remove terms that have specific digits in it, like 9, suddenly it doesn't diverge anymore.

But I like Truncated Harmonic Distribution. Once someone finds out an existing name for it, it will become the Distribution Formerly Known as the Truncated Harmonic Distribution, of course. :)