What is an example of two (or more) not independent but identically distributed random variables?
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Let $X,\,Z$ denote independent and identically distributed variables, let $W$ be a (not necessarily fair) coin toss, and let $Y$ be either $X$ or $Z$ based on $W$. Then $X,\,Y$ are identically distributed but not independent.
Take the random variable $X$, with your favourite (non-degenerate) distribution, and the random variable $Y=X$. These two are very much dependent and identically distributed.
For a less trivial example, if the distribution of $X$ is synetic about $0$ (say uniform on $[-1,1]$ or standard normal), you can use $Y=-X$.
Another example: Let $X$ be the number of pips on the top face of a fair six-sided die that is rolled. Let $Y$ be the number of pips on the bottom face of the same die.
For a very simple and intuitive example just take
Think of it as a fair game where $X$ and $Y$ reflect players competing against each other. A $1$ corresponds to winning and a $0$ to losing. Both players have the same chance of winning, but the outcome for $X$ and $Y$ is not independent at all.