That is in-fact the typical target of these laws, but that is because it will result in a fairly good approximation.
The problem arises when you try to hit your chosen distribution.
Random process often produce uniform distributions, but uniform distributions do not confirm something is random. It is just a datapoint that something may be random.
As Cronos said you could have other curves you could choose, but by definition PRNGs are not random and are 100% fully deterministic. The intent is to emulate randomness, but in trying to fit that curve you can actually induce patterns by overfitting the curve.
While I am flattered that you may assume that I possess the power, I have never sat on a gaming commission or a state or federal gaming control board.
I was describing a typical chi-squared test for goodness of fit test that is typically codified through law or regulation. It is a legal requirement and not a design decision.
Wouldn’t the p-values from a goodness-of-fit test be uniformly distributed between 0 and 1, assuming you are indeed dealing with a random source? I still do not see how the normal distribution comes into it. (Other than the fact that summing the squares of normal random variables results in a chi-squared distribution.)
With an infinite sized data source is should tend to a normal distribution, but in a true random there is no requirement for it to approach a normal distribution.
This goes back to the monte carlo fallacy. If you flip a fair coin 10 times in a row and it lands heads 10 times the next toss probabilities are in no way influenced by that extremely rare event of getting 10 heads in a row.
On that 11th coin toss, it has the exact same odds it had in the first coin toss. This type of noise breaks chi-squared tests and by law you have to meet the chi-squared test.
The assumption that true random process will result in a normalized curve is where this line of thinking is wrong.
A chi-squared distribution in these laws is typically the simplest form or the square of a standard normal distribution. A chi-squared test can be used when you expect a normal distribution. It is simply the case of making legal products involves targeting a normal distribution.
Deja vu all over again?. The upshot from 2005 was a better description was ‘unpredictable,’ not specifically ‘random,’ because then you start getting bogged down in all the PRNG stuff that you’re seeing.
I think you are confusing the distribution of the outcomes with the distribution of the sample means. The first can be any distribution whereas the later is approximately normal (Central Limit Theorem)
I am afraid I still do not follow. Could you elaborate by giving precise details of how the state commission runs the test? Does the methodology vary when testing purely mechanical machines (if any exist), roulette wheels, and such?
Here is my intuitive reasoning: suppose your test statistic is X. Then the p-value of your test is defined to be the probability, under your randomness assumptions, that X would take on a value at least as extreme as that which you observed in your test. E.g., if X has the cumulative distribution function F(x) = Prob[X ≤ x], then p = Pr[X ≥ x], where x is your observed value. Assume that F is continuous; then p = 1 - F(x) and Prob[1 - F(X) ≤ p] = Prob[F(X) ≥ 1-p] = Prob[X ≥ F[sup]-1/sup] = 1 - Prob[X < F[sup]-1/sup] = 1 - F(F[sup]-1/sup) = p, so p is uniformly distributed in this case.
So if you run the test lots and lots of times, the resulting p-values in this case tend to a uniform distribution. It seems that if you observe a normal distribution (and how could you, considering that p is constrained between 0 and 1), something is horribly wrong!
No I am simplifying for a platform that doesn’t support math, and with an audience which wouldn’t work through it anyway.
Feel free to show how ‘chi-square test for goodness of fit’ is not sensitive to noise, or that isn’t the test that an electronic product needs to be developed to to meet the regulations.
There is little value in arguing terminology, the fact is still that products have to be produced to match the test, as required by law and or regulation.
