Doesn't *identically distributed* imply *independent*?

Question

What the title says. If

• I draw a random value $x_1 \sim \mathcal{N}(\mu, \sigma)$
• a minute later, I draw another $x_2 \sim \mathcal{N}(\mu, \sigma)$

they come from identical distributions. Is there any scenario in which that does not necessarily imply that they are independent?

Answer 1

Here's a simple example. Roll a die, to give a random number $N$. Let $X$ be the variable $\lfloor N/4\rfloor$, and $Y$ be the random variable $N\pmod 2$.

$X$ and $Y$ are identically distributed $\mathrm{Bernoulli}(1/2)$ random variables, but they are not independent: $\Pr(X=Y=1)=1/6$.

Answer 2

If e.g. $\mu=0$ in your example then $x_1$ and $-x_1$ are identically distributed but are not independent.