Best Of

You can still do the chi-square test, you just need to only do it over the domain of the random variable (that is, only over the numbers that have a 9 in them)

For guesswork I think this would help a lot:

http://gamedev.stackexchange.com/questions/12638/weighted-random-distribution

On that note, when it comes to discrete simulations you can certainly generate values randomly with distribution A and then use that value through a second distribution B to generate new values. However, I can't possibly think how to subtract one distribution from another.

Knowing nothing else about the problem at hand, I think you will have to speculate to the best of your ability because reverse engineering could very well be impossible. At least to my knowledge. If you ask a math PhD, they are going to ask you for the data and say give me a week.

By definition a uniform or rectangular distribution gives the exact same odds of any number to be generated.
So if we take a step back and simplify this then there are one of three scenarios I can think of:
1. This is not a uniform distribution
2. Your data set is too small and even though it appears to not be uniform, it is not
3. The actual programmatic generator of this data is altering the outcome /after/ the number is generated to avoid certain outcomes.

Even if I knew the nature of the problem at hand, it wouldn't be prudent to speculate.

This is the problem with reverse engineering. Even if you find a distribution and simulate it to give you very similar results, you won't know exactly what is happening unless you can peak at the source.

The data is generated by a very strong random number generator that has already passed a lot of statistical tests. This is then used to generate random digits 0..9. A ChiSq on the individual digits confirms the distribution appears uniform.

The second part of the problem is to look at the distribution individual digits in larger numbers 0..N for some large N where there is at least one '9'. Things like mean and variance look good but a ChiSq of this rejects uniformity as you would expect given that some values are being discarded. The problem is really to model the expected distribution when N is large.

Not sure if there is a way to model this, or to subtract one distribution from another, or add several distributions together, or if there is another test that would be better.

Chi Square is very useful because it predicts the commonality between a theoretical model and actual data. I'm a tad confused if he has a model or just data. But let's set the Chi Square distribution aside.

I /think/ what he is saying through you is that he only has a raw set of data, and he wants to see if he can reverse engineer it in to a distribution. I did this type of thing back when I had to create simulations. This is often times guess work based on the source or nature of the data since there are so many distrubutions used with random numbers (normal, poisson, gamma, lognormal, whatever). The way I would then prove it is by simulating random number generation through the distribution I guessed it was, and then analyze the difference between the simulated numbers and the original data. That's how you test the accuracy of a simulation with known outcomes.

Can you, or he via you, give us an idea of where the raw data comes from and what exactly it represents? With that I could give you a list of distributions the numbers are likely to have been generated from.