Experimentially proving that random variables are independent How would I, experimentally prove that two random variables drawn either from the same population or two different populations are independent (or for that matter of fact dependent).
 A: @Procrastinator has offered good tools to check a possible relationship.
If you really want to get mathematical, get ready because this is actually a very very deep problem. You should first estimate their dependency by some sort of correlation estimator. Then, you'll do inference on that estimate. That is, you will do a hypothesis test to see if that correlation is equal to zero or not.
I can think of two things:
1) Assume that they come from a multivariate normal distribution. Then, they are independent if and only if Pearson's correlation is equal to zero. However, know that testing correlation is a bitch. You will really need a large sample size for the test. Look at the graph here to get some ideas.
Caveat: Even if sample 1 and sample 2 has a normal distribution, this doesn't mean that they come from a multivariate normal distribution. Someone else might elaborate more on this.
2) This question was bothering Gabor Szekely for a long long time. He recently come up with the solution: distance correlation. Again, you will have to estimate the distance correlation from your sample. But the good thing is that, distance correlation is actually equal to zero if your samples are actually independent. This helps you to get around the multivariate normal assumption. You should estimate the distance correlation from the sample, and then do hypothesis testing as you would have done with a multivariate normal.
