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Identically distributed implies that each random variable has the same probability distribution as the others. Independent means that all the random variables are mutually independent. The notion of dependent but identically distributed random variables is used in statistical literature (for example here).

However, if the variables are not independent, then the distribution of one variable is affected by other variables. If this is the case, then how can they be identically distributed? It seems like random variable cannot be dependent and identically distributed at the same time.

What am I getting wrong?

Edit: After looking at examples in the comments, I realized I did not ask the right question. I meant a sample, and not random variables. We known a sample can be drawn i.i.d from a distribution. What does it mean for a sample to be identically distributed but not independent.

amitdatta
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    Take two random variables $X_1$ and $X_2$ that are, for example, bivariate normal with $\mu_1=\mu_2$ and $\sigma_1=\sigma_2$ as well as some correlation $\rho\neq0$. They have the same marginal distributions but are not independent. – Christoph Hanck Sep 21 '16 at 14:30
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    Another example is $X\sim$ Bernoulli$(0.5)$ and $Y = 1-X$. – Michael M Sep 21 '16 at 14:34
  • An example in a sampling setting is to fill an urn with balls. Create a sample of size $n$ by drawing a *single* ball from the urn and creating $n-1$ copies of that ball. Obviously the copies are identically distributed but, even more obviously, they are not independent. – whuber Sep 21 '16 at 15:39

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