# Independent but not identically distributed

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 ?

• 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. – Glen_b Oct 14 '13 at 3:31
• 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? – gung - Reinstate Monica Oct 14 '13 at 3:34
• More details/context might help – Glen_b Oct 14 '13 at 3:41
• @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. – Daniel Wonglee Oct 14 '13 at 5:31