# Identically distributed and not independent random variables [closed]

What is an example of two (or more) not independent but identically distributed random variables?

• Say $X$ is uniformly distributed on $[-1,1]$ and $Y=-X$. – David C. Ullrich Jul 22 at 15:47
• Another easy question that everyone wants to answer, but no one sees that its quality is debatable – Jakobian Jul 22 at 15:52

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

$$P(X=1)=P(X=0)=\frac12$$

and

$$Y=1-X.$$

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.