Are random selections from i.i.d. random variables independent? Let us have identically independently distributed random variables $x_1, x_2, \dots, x_{10}$. Now let us pick indices $\alpha, \beta$ uniformly independently from $1,2,\dots,10$. Are variables $x_{\alpha}$ and $x_{\beta}$ independent?
My intuition says that they are not, since there is a chance $\frac{1}{10}$ that they are the same. But how can I formally prove this? (or prove that they are independent in case I am wrong)

To summarize:

With replacement (the question), $(\alpha,\beta)$ is independent and $(x_\alpha,x_\beta)$ is not. Without replacement, $(\alpha,\beta)$ is not independent and $(x_\alpha,x_\beta)$ is.

To show the case with replacement, consider
$$P(x_\alpha\in A,x_\beta\in A)=P(x_\alpha\in A,x_\beta\in A,\alpha\ne\beta)+P(x_\alpha\in A,\alpha=\beta),
$$
thus,
$$
P(x_\alpha\in A,x_\beta\in A)=\left(1-\frac1{10}\right)p^2+\frac1{10}p,
$$
where $p=P(x_1\in A)$. On the other hand,
$$
P(x_\alpha\in A)P(x_\beta\in A)=p^2.
$$
If $p$ is not $0$ or $1$ these are different.
