Let $X_1, X_2,\ldots ,X_n$ be discrete random variables.

I'm looking for a way to prove the random variables are independent but not identically distributed.

Can anyone suggest some ideas ?

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    $\begingroup$ You can't prove independence from a sample. You might find that your data are consistent with independence, but they'd also be consistent with mild dependence. Showing that they're inconsistent with being iid should be easier. $\endgroup$ – Glen_b Oct 14 '13 at 3:31
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    $\begingroup$ In what sense do you want a proof? Are you just trying to understand the ideas? Is this a class assignment? What would having such a proof help you achieve? $\endgroup$ – gung - Reinstate Monica Oct 14 '13 at 3:34
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    $\begingroup$ More details/context might help $\endgroup$ – Glen_b Oct 14 '13 at 3:41
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    $\begingroup$ @gung: I'm working on a machine learning problem. When I assumed the data is independent but not identically distributed, I got better results than assuming IID. Hence I would like to prove the data is independent but not identically distributed. $\endgroup$ – Daniel Wonglee Oct 14 '13 at 5:31

First things first. There needs to be greater information given as this does not have a universally correct answer. Different types of distributions have to be looked at with different types of procedures.

But just to show that yes this is possible, we assume that each of the variables that you have mentioned are normally distributed but the parameters of the normal distributions are different from each other for any given pair.

Now we take n samples each of these variables. Then calculate the correlation coefficients for each pair of the variables. If we cannot reject the hypothesis of these correlation coefficients being zero, we hypothesize that the variables are independent of each other. So we have a set of variables which are independent from each other, but they have different probability distributions.

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  • $\begingroup$ Thank you for your reply. However I don't understand 'We group together each of the variables separately and then calculate the correlation coefficients for each pair of the groups'. Can you please explain it ? $\endgroup$ – Daniel Wonglee Oct 14 '13 at 6:28
  • $\begingroup$ have edited the answer, please do read it again. sorry for the earlier ambiguity. $\endgroup$ – htrahdis Oct 14 '13 at 7:13
  • $\begingroup$ May I know the reason behind saying that they have different probability distributions if we cannot reject the hypothesis of these correlation coefficients being zero? $\endgroup$ – Daniel Wonglee Oct 14 '13 at 7:49
  • $\begingroup$ This would be the definition of random variables which supposed to be independend (this only holds for the normal distribution or spheric distributions) but not iid. $\endgroup$ – Druss2k Oct 14 '13 at 7:55
  • $\begingroup$ i have said that different cases are to be handled separately and this is just one particular case. and different normally distributed random variables having different parameters of normal distribution are said to be of different distributions if that is the confusion here. $\endgroup$ – htrahdis Oct 16 '13 at 13:39

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