For a geometric distribution, I know there's a certain pattern, for example, given the probability of success $0.2$ for a event, you can have:
$$P(X = r) = {q}^{(r-1)} \times p$$
- When r = 1, probability is 0.2
- When r = 2, probability is 0.32
- When r = 3, probability is 0.128
- So on..
This tells me the probability of trial 1, 2, 3 .. and I can see it decreases for every trial.
I tried to do this same example using code to generate data for geometric distribution using python programming language:
s = np.random.geometric(p, 100000)
It means for 'p' = 0.20, give me 100,000 samples.
Here are 500 samples from these and what they look like:
[1, 6, 1, 2, 3, 2, 3, 1, 5, 3, 9, 12, 6, 2, 7, 3, 7, 7, 4, 2, 4, 4, 1, 2, 9, 3, 9, 1, 5, 9, 1, 10, 7, 8, 2, 1, 6, 1, 8, 1, 5, 2, 2, 7, 4, 3, 7, 3, 6, 4, 5, 4, 1, 9, 1, 7, 6, 2, 7, 13, 24, 1, 3, 2, 2, 8, 2, 1, 2, 8, 3, 2, 1, 2, 3, 12, 7, 5, 14, 6, 2, 3, 1, 1, 4, 3, 1, 4, 1, 12, 2, 7, 18, 8, 6, 5, 5, 6, 2, 12, 4, 1, 1, 2, 5, 3, 2, 2, 2, 4, 3, 12, 6, 7, 5, 9, 3, 12, 3, 5, 9, 7, 3, 8, 3, 12, 2, 27, 2, 7, 2, 14, 3, 2, 5, 5, 9, 1, 1, 12, 6, 1, 12, 14, 4, 2, 7, 25, 7, 1, 3, 1, 3, 3, 10, 2, 4, 1, 6, 11, 2, 12, 5, 5, 14, 3, 1, 1, 1, 2, 1, 10, 8, 6, 4, 1, 2, 2, 3, 14, 13, 2, 3, 2, 8, 8, 5, 6, 2, 6, 1, 5, 2, 12, 1, 6, 1, 3, 12, 9, 6, 4, 1, 4, 6, 16, 8, 3, 3, 4, 1, 7, 2, 1, 1, 4, 4, 7, 2, 5, 3, 7, 8, 6, 3, 7, 2, 8, 2, 1, 1, 4, 9, 2, 5, 4, 2, 10, 1, 10, 6, 10, 1, 6, 2, 3, 14, 2, 1, 10, 4, 2, 11, 6, 10, 6, 1, 3, 3, 4, 2, 1, 1, 1, 2, 9, 19, 1, 2, 17, 1, 6, 7, 12, 4, 15, 11, 1, 3, 3, 4, 2, 31, 2, 22, 6, 2, 4, 10, 1, 1, 3, 9, 3, 4, 4, 15, 8, 5, 1, 1, 18, 2, 3, 2, 1, 11, 5, 8, 8, 2, 8, 4, 7, 3, 2, 5, 1, 8, 2, 20, 1, 8, 2, 1, 4, 4, 3, 5, 2, 5, 2, 1, 1, 9, 3, 6, 11, 2, 5, 7, 1, 1, 10, 1, 1, 1, 3, 7, 2, 6, 4, 4, 9, 3, 1, 3, 3, 1, 1, 7, 7, 1, 3, 2, 2, 21, 4, 1, 12, 7, 4, 7, 5, 3, 4, 1, 2, 4, 2, 3, 1, 16, 21, 7, 2, 9, 3, 4, 2, 6, 9, 2, 1, 1, 32, 5, 3, 5, 2, 5, 4, 5, 1, 2, 7, 4, 18, 12, 3, 2, 4, 12, 10, 5, 8, 3, 1, 7, 7, 7, 8, 9, 10, 1, 22, 7, 2, 16, 5, 9, 1, 2, 2, 2, 4, 1, 3, 7, 7, 4, 14, 9, 3, 4, 4, 5, 4, 1, 7, 3, 1, 4, 5, 3, 2, 19, 2, 2, 17, 4, 4, 5, 3, 9, 6, 3, 5, 5, 2, 6, 1, 8, 6, 1, 1, 2, 6, 11, 3, 3, 6, 9, 4, 8, 1, 9, 6, 3, 8, 9, 5, 11, 2, 1, 3, 1, 6, 3, 1]
What I want to know is that how these numbers represent geometric distribution? Is there a certain order? The probability of say, 0, 1, 2, or 3 appearing as next number? Does the order matter at all?
Looking at your data, you generated random numbers from a geometric distribution counting the failures before first success
$$P(X=x)=0.8^x\cdot 0.2$$
$x=0;1;2...$
If you do the average of the generated numbers you have to find 4 and a variance of 20. Thus it is difficult that your sample comes from a $Geo(0.2)$
No, there is no order, correlation or dependence in those numbers (I hope, if the random generating algorithm works well)
You can use particular statistical methods to generate this sample by your own