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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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.